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| author | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
|---|---|---|
| committer | Cyril Cohen | 2019-10-16 11:26:43 +0200 |
| commit | 6b59540a2460633df4e3d8347cb4dfe2fb3a3afb (patch) | |
| tree | 1239c1d5553d51a7d73f2f8b465f6a23178ff8a0 /docs/htmldoc/mathcomp.solvable.maximal.html | |
| parent | dd82aaeae7e9478efc178ce8430986649555b032 (diff) | |
removing everything but index which redirects to the new page
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diff --git a/docs/htmldoc/mathcomp.solvable.maximal.html b/docs/htmldoc/mathcomp.solvable.maximal.html deleted file mode 100644 index ba5d392..0000000 --- a/docs/htmldoc/mathcomp.solvable.maximal.html +++ /dev/null @@ -1,805 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" -"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml"> -<head> -<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> -<link href="coqdoc.css" rel="stylesheet" type="text/css" /> -<title>mathcomp.solvable.maximal</title> -</head> - -<body> - -<div id="page"> - -<div id="header"> -</div> - -<div id="main"> - -<h1 class="libtitle">Library mathcomp.solvable.maximal</h1> - -<div class="code"> -<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> - Distributed under the terms of CeCILL-B. *)</span><br/> - -<br/> -</div> - -<div class="doc"> - This file establishes basic properties of several important classes of - maximal subgroups: maximal, max and min normal, simple, characteristically - simple subgroups, the Frattini and Fitting subgroups, the Thompson - critical subgroup, special and extra-special groups, and self-centralising - normal (SCN) subgroups. In detail, we define: - charsimple G == G is characteristically simple (it has no nontrivial - characteristic subgroups, and is nontrivial) - 'Phi(G) == the Frattini subgroup of G, i.e., the intersection of - all its maximal proper subgroups. - 'F(G) == the Fitting subgroup of G, i.e., the largest normal - nilpotent subgroup of G (defined as the (direct) - product of all the p-cores of G). - critical C G == C is a critical subgroup of G: C is characteristic - (but not functorial) in G, the center of C contains - both its Frattini subgroup and the commutator [G, C], - and is equal to the centraliser of C in G. The - Thompson_critical theorem provides critical subgroups - for p-groups; we also show that in this case the - centraliser of C in Aut G is a p-group as well. - special G == G is a special group: its center, Frattini, and - derived sugroups coincide (we follow Aschbacher in - not considering nontrivial elementary abelian groups - as special); we show that a p-group factors under - coprime action into special groups (Aschbacher 24.7). - extraspecial G == G is a special group whose center has prime order - (hence G is non-abelian). - 'SCN(G) == the set of self-centralising normal abelian subgroups - of G (the A <| G such that 'C_G(A) = A). - 'SCN_n(G) == the subset of 'SCN(G) containing all groups with rank - at least n (i.e., A \in 'SCN(G) and 'm(A) >= n). -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> - -<br/> -<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Defs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="charsimple"><span class="id" title="definition">charsimple</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">min</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">of</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#83949b4ab4a5556fc861a975723584d5"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Frattini"><span class="id" title="definition">Frattini</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">bigcap_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">(</span></a><span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#a7a783aaf5f1e41d47b478dd81a17722"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Frattini_group</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini"><span class="id" title="definition">Frattini</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f6996ff347e6cf832aa130837b06a848"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="Fitting"><span class="id" title="definition">Fitting</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">p</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#primes"><span class="id" title="definition">primes</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_group_set"><span class="id" title="lemma">Fitting_group_set</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Fitting"><span class="id" title="definition">Fitting</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fitting_group</span> <span class="id" title="var">G</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Fitting_group_set"><span class="id" title="lemma">Fitting_group_set</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="critical"><span class="id" title="definition">critical</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini"><span class="id" title="definition">Frattini</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="special"><span class="id" title="definition">special</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.solvable.maximal.html#Frattini"><span class="id" title="definition">Frattini</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="extraspecial"><span class="id" title="definition">extraspecial</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="SCN"><span class="id" title="definition">SCN</span></a> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">set</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#3e9d12914c7607378976c499a50a47c7"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Definition</span> <a name="SCN_at"><span class="id" title="definition">SCN_at</span></a> <span class="id" title="var">n</span> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">set</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN"><span class="id" title="definition">SCN</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#89384e246d9189b85a2e3f87a816b040"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#89384e246d9189b85a2e3f87a816b040"><span class="id" title="notation">r</span></a><a class="idref" href="mathcomp.solvable.abelian.html#89384e246d9189b85a2e3f87a816b040"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.abelian.html#89384e246d9189b85a2e3f87a816b040"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#91816551bcea1b6f359ecf76f3595e38"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Defs"><span class="id" title="section">Defs</span></a>.<br/> - -<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">"</span></a>''Phi' ( A )" := (<a class="idref" href="mathcomp.solvable.maximal.html#Frattini"><span class="id" title="definition">Frattini</span></a> <span class="id" title="var">A</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''Phi' ( A )") : <span class="id" title="var">group_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="8ee60567bf9d95c620ba0d9e846ea5ad"><span class="id" title="notation">"</span></a>''Phi' ( G )" := (<a class="idref" href="mathcomp.solvable.maximal.html#Frattini_group"><span class="id" title="definition">Frattini_group</span></a> <span class="id" title="var">G</span>) : <span class="id" title="var">Group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">"</span></a>''F' ( G )" := (<a class="idref" href="mathcomp.solvable.maximal.html#Fitting"><span class="id" title="definition">Fitting</span></a> <span class="id" title="var">G</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''F' ( G )") : <span class="id" title="var">group_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="703dbbbe1d72723e0ed1ecbc9c3c00a4"><span class="id" title="notation">"</span></a>''F' ( G )" := (<a class="idref" href="mathcomp.solvable.maximal.html#Fitting_group"><span class="id" title="definition">Fitting_group</span></a> <span class="id" title="var">G</span>) : <span class="id" title="var">Group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Notation</span> <a name="d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">"</span></a>''SCN' ( B )" := (<a class="idref" href="mathcomp.solvable.maximal.html#SCN"><span class="id" title="definition">SCN</span></a> <span class="id" title="var">B</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''SCN' ( B )") : <span class="id" title="var">group_scope</span>.<br/> -<span class="id" title="keyword">Notation</span> <a name="cdedee86bae1183b22902f09a64cf7f5"><span class="id" title="notation">"</span></a>''SCN_' n ( B )" := (<a class="idref" href="mathcomp.solvable.maximal.html#SCN_at"><span class="id" title="definition">SCN_at</span></a> <span class="id" title="var">n</span> <span class="id" title="var">B</span>)<br/> - (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''SCN_' n ( B )") : <span class="id" title="var">group_scope</span>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="PMax"><span class="id" title="section">PMax</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="PMax.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="PMax.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="PMax.P"><span class="id" title="variable">P</span></a> <a name="PMax.M"><span class="id" title="variable">M</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="PMax.pP"><span class="id" title="variable">pP</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#PMax.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_maximal_normal"><span class="id" title="lemma">p_maximal_normal</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_maximal_index"><span class="id" title="lemma">p_maximal_index</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_index_maximal"><span class="id" title="lemma">p_index_maximal</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#PMax.P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#PMax"><span class="id" title="section">PMax</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Frattini"><span class="id" title="section">Frattini</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="Frattini.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> <span class="id" title="var">M</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_sub"><span class="id" title="lemma">Phi_sub</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_sub_max"><span class="id" title="lemma">Phi_sub_max</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#M"><span class="id" title="variable">M</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_proper"><span class="id" title="lemma">Phi_proper</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f8a5284fd4c86e7670efa54241b49929"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_nongen"><span class="id" title="lemma">Phi_nongen</span></a> <span class="id" title="var">G</span> <span class="id" title="var">X</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0d7ccd69af81527d9facc6293603bbef"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#X"><span class="id" title="variable">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.solvable.maximal.html#X"><span class="id" title="variable">X</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Frattini_continuous"><span class="id" title="lemma">Frattini_continuous</span></a> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) <span class="id" title="var">G</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini"><span class="id" title="section">Frattini</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Frattini_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Phi_sub"><span class="id" title="lemma">Phi_sub</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini_continuous"><span class="id" title="lemma">Frattini_continuous</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Frattini_gFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini_continuous"><span class="id" title="lemma">Frattini_continuous</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Frattini0"><span class="id" title="section">Frattini0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Frattini0.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">D</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini0.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_char"><span class="id" title="lemma">Phi_char</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_normal"><span class="id" title="lemma">Phi_normal</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_Phi"><span class="id" title="lemma">injm_Phi</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">D</span> <span class="id" title="var">G</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_Phi"><span class="id" title="lemma">isog_Phi</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">G</span> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="PhiJ"><span class="id" title="lemma">PhiJ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini0"><span class="id" title="section">Frattini0</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Frattini2"><span class="id" title="section">Frattini2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="Frattini2.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_quotient_id"><span class="id" title="lemma">Phi_quotient_id</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_quotient_cyclic"><span class="id" title="lemma">Phi_quotient_cyclic</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#cyclic"><span class="id" title="definition">cyclic</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Frattini2.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Frattini2.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trivg_Phi"><span class="id" title="lemma">trivg_Phi</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2.P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini2"><span class="id" title="section">Frattini2</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Frattini3"><span class="id" title="section">Frattini3</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Frattini3.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Frattini3.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Frattini3.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="Frattini3.pP"><span class="id" title="variable">pP</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_quotient_abelem"><span class="id" title="lemma">Phi_quotient_abelem</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_joing"><span class="id" title="lemma">Phi_joing</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0d7ccd69af81527d9facc6293603bbef"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">Mho</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">^</span></a>1<a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_Mho"><span class="id" title="lemma">Phi_Mho</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">Mho</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">^</span></a>1<a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Frattini3.P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.abelian.html#2fc9e45ba06c586f77f5571683771ca7"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini3"><span class="id" title="section">Frattini3</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Frattini4"><span class="id" title="section">Frattini4</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Frattini4.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Frattini4.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">P</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="PhiS"><span class="id" title="lemma">PhiS</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_Phi"><span class="id" title="lemma">morphim_Phi</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">P</span> <span class="id" title="var">D</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="quotient_Phi"><span class="id" title="lemma">quotient_Phi</span></a> <span class="id" title="var">P</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher (23.2) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Phi_min"><span class="id" title="lemma">Phi_min</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_cprod"><span class="id" title="lemma">Phi_cprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Phi_mulg"><span class="id" title="lemma">Phi_mulg</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#8b8794efbfbae1b793d9cb62ce802285"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">Phi</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.solvable.maximal.html#28e1060c98ad15195c81f60c91d64b08"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="charsimpleP"><span class="id" title="lemma">charsimpleP</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">K</span>, <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>)<br/> - (<a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Frattini4"><span class="id" title="section">Frattini4</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Fitting"><span class="id" title="section">Fitting</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Fitting.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Fitting.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_normal"><span class="id" title="lemma">Fitting_normal</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_sub"><span class="id" title="lemma">Fitting_sub</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_nil"><span class="id" title="lemma">Fitting_nil</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_max"><span class="id" title="lemma">Fitting_max</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="pcore_Fitting"><span class="id" title="lemma">pcore_Fitting</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_core_Fitting"><span class="id" title="lemma">p_core_Fitting</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="nilpotent_Fitting"><span class="id" title="lemma">nilpotent_Fitting</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#nilpotent"><span class="id" title="definition">nilpotent</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_eq_pcore"><span class="id" title="lemma">Fitting_eq_pcore</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#ca29ecf9a3780bf15fe608e2d2c00594"><span class="id" title="notation">^'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="FittingEgen"><span class="id" title="lemma">FittingEgen</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation">bigcup_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation">(</span></a><span class="id" title="var">p</span> <a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#aed478b27f23b4f753c27c8ac393febc"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">pi</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#c36dd927e8fe3f2052f45795266a50d2"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#044559fabfce2725f5d4a93e8fca2034"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7fae3285ae0b14be2ed14c5a340c6d3d"><span class="id" title="notation">>></span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Fitting"><span class="id" title="section">Fitting</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="FittingFun"><span class="id" title="section">FittingFun</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="morphim_Fitting"><span class="id" title="lemma">morphim_Fitting</span></a> : <a class="idref" href="mathcomp.solvable.gfunctor.html#GFunctor.pcontinuous"><span class="id" title="definition">GFunctor.pcontinuous</span></a> (@<a class="idref" href="mathcomp.solvable.maximal.html#Fitting"><span class="id" title="definition">Fitting</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="FittingS"><span class="id" title="lemma">FittingS</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="FittingJ"><span class="id" title="lemma">FittingJ</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#049e6d4210dc2b8af76facf30c9d4dd6"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#FittingFun"><span class="id" title="section">FittingFun</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fitting_igFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">igFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Fitting_sub"><span class="id" title="lemma">Fitting_sub</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#morphim_Fitting"><span class="id" title="lemma">morphim_Fitting</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#c0205c751a17b7793ccdaf02cc4999e3"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fitting_gFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">gFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#morphim_Fitting"><span class="id" title="lemma">morphim_Fitting</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#cecbde1597e0d77a491e8c4f94033af4"><span class="id" title="notation">]</span></a>.<br/> -<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Fitting_pgFun</span> := <a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">pgFun</span></a> <a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#morphim_Fitting"><span class="id" title="lemma">morphim_Fitting</span></a><a class="idref" href="mathcomp.solvable.gfunctor.html#f4a0ec7c18bd128b271d4428328fd43b"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="IsoFitting"><span class="id" title="section">IsoFitting</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="IsoFitting.gT"><span class="id" title="variable">gT</span></a> <a name="IsoFitting.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="IsoFitting.G"><span class="id" title="variable">G</span></a> <a name="IsoFitting.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<a name="IsoFitting.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_char"><span class="id" title="lemma">Fitting_char</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a>. <br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_Fitting"><span class="id" title="lemma">injm_Fitting</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_Fitting"><span class="id" title="lemma">isog_Fitting</span></a> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#IsoFitting"><span class="id" title="section">IsoFitting</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="CharSimple"><span class="id" title="section">CharSimple</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="CharSimple.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">L</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#CharSimple.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">p</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="minnormal_charsimple"><span class="id" title="lemma">minnormal_charsimple</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="maxnormal_charsimple"><span class="id" title="lemma">maxnormal_charsimple</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">L</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="abelem_split_dprod"><span class="id" title="lemma">abelem_split_dprod</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p_abelem_split1"><span class="id" title="lemma">p_abelem_split1</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">p</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2242f6721707980eca939ec29164eab3"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#09508509373ff3217fbbadb14d25dc5c"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#5e5825d099c952c2db2842c142cbde94"><span class="id" title="notation">]></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#191b5570f070a51bd5c860222c206828"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="abelem_charsimple"><span class="id" title="lemma">abelem_charsimple</span></a> <span class="id" title="var">p</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="charsimple_dprod"><span class="id" title="lemma">charsimple_dprod</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#CharSimple.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">I</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.perm.html#683bed2e43e8b9b52492c89e618e34b6"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.perm.html#683bed2e43e8b9b52492c89e618e34b6"><span class="id" title="notation">perm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#CharSimple.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.perm.html#683bed2e43e8b9b52492c89e618e34b6"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8708f36d374a98f4d683c7593d1ea6a"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#59ba2b47d2814e66f8210a649ae6e6bc"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">(</span></a><span class="id" title="var">f</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#I"><span class="id" title="variable">I</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#d37140b0b5d9683da109df6bc7f32772"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#de2c3fcab69008133cce8f8fc06f2b4b"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="simple_sol_prime"><span class="id" title="lemma">simple_sol_prime</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="charsimple_solvable"><span class="id" title="lemma">charsimple_solvable</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#charsimple"><span class="id" title="definition">charsimple</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#is_abelem"><span class="id" title="definition">is_abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="minnormal_solvable"><span class="id" title="lemma">minnormal_solvable</span></a> <span class="id" title="var">L</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#is_abelem"><span class="id" title="definition">is_abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#d7e433f5d2fe56f5b712860a9ff2a681"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="solvable_norm_abelem"><span class="id" title="lemma">solvable_norm_abelem</span></a> <span class="id" title="var">L</span> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#CharSimple.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#L"><span class="id" title="variable">L</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#is_abelem"><span class="id" title="definition">is_abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#554fc3f3cf0a27fe0863b7741d119014"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="trivg_Fitting"><span class="id" title="lemma">trivg_Fitting</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b91223a7636398c530555b2312d1e79b"><span class="id" title="notation">:==:</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Fitting_pcore"><span class="id" title="lemma">Fitting_pcore</span></a> <span class="id" title="var">pi</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">O_pi</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">F</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#23baa6894d2fc69111d3059c4d7a7bd5"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#64e6c9ddce70097e2cb88e9baa3b5a39"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#CharSimple"><span class="id" title="section">CharSimple</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="SolvablePrimeFactor"><span class="id" title="section">SolvablePrimeFactor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="SolvablePrimeFactor.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="index_maxnormal_sol_prime"><span class="id" title="lemma">index_maxnormal_sol_prime</span></a> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="sol_prime_factor_exists"><span class="id" title="lemma">sol_prime_factor_exists</span></a> :<br/> - <a class="idref" href="mathcomp.solvable.nilpotent.html#solvable"><span class="id" title="definition">solvable</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#be2f022a539ec6d4d51932b5ea998e57"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">{</span></a><span class="id" title="var">H</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#f92718946b2f68c8f7100be4d6b45f82"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#SolvablePrimeFactor"><span class="id" title="section">SolvablePrimeFactor</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Special"><span class="id" title="section">Special</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Special.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="Special.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Special.A"><span class="id" title="variable">A</span></a> <a name="Special.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher (23.7) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="center_special_abelem"><span class="id" title="lemma">center_special_abelem</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Special.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="exponent_special"><span class="id" title="lemma">exponent_special</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Special.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#exponent"><span class="id" title="definition">exponent</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2.<br/> - -<br/> -</div> - -<div class="doc"> - Aschbacher 24.7 (replaces Gorenstein 5.3.7) -</div> -<div class="code"> -<span class="id" title="keyword">Theorem</span> <a name="abelian_charsimple_special"><span class="id" title="lemma">abelian_charsimple_special</span></a> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Special.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">bigcup_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">(</span></a><span class="id" title="var">H</span> <a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#d9dc63f0c53bc5e6f232c50d48c40709"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#ec4dd368465f0599cdb8c6e3313efc95"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Special.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Special"><span class="id" title="section">Special</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Extraspecial"><span class="id" title="section">Extraspecial</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="Extraspecial.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="Extraspecial.gT"><span class="id" title="variable">gT</span></a> <a name="Extraspecial.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">D</span> <span class="id" title="var">E</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">M</span> <span class="id" title="var">R</span> <span class="id" title="var">S</span> <span class="id" title="var">T</span> <span class="id" title="var">U</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Extraspecial.Basic"><span class="id" title="section">Basic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Extraspecial.Basic.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypotheses</span> (<a name="Extraspecial.Basic.pS"><span class="id" title="variable">pS</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a>) (<a name="Extraspecial.Basic.esS"><span class="id" title="variable">esS</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.Basic.pZ"><span class="id" title="variable">pZ</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.solvable.pgroup.html#pgroupS"><span class="id" title="lemma">pgroupS</span></a> (<a class="idref" href="mathcomp.solvable.center.html#center_sub"><span class="id" title="lemma">center_sub</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.pS"><span class="id" title="variable">pS</span></a>.<br/> -<span class="id" title="keyword">Lemma</span> <a name="extraspecial_prime"><span class="id" title="lemma">extraspecial_prime</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_center_extraspecial"><span class="id" title="lemma">card_center_extraspecial</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="min_card_extraspecial"><span class="id" title="lemma">min_card_extraspecial</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#73030c22bc0b1fa771c65aa5414c65f9"><span class="id" title="notation">≥</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 3.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.Basic"><span class="id" title="section">Basic</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="card_p3group_extraspecial"><span class="id" title="lemma">card_p3group_extraspecial</span></a> <span class="id" title="var">E</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 3)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="p3group_extraspecial"><span class="id" title="lemma">p3group_extraspecial</span></a> <span class="id" title="var">G</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 3 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="extraspecial_nonabelian"><span class="id" title="lemma">extraspecial_nonabelian</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b3ebd0deddd84fd60e149cb5ef719351"><span class="id" title="notation">~~</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="exponent_2extraspecial"><span class="id" title="lemma">exponent_2extraspecial</span></a> <span class="id" title="var">G</span> : 2<a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#exponent"><span class="id" title="definition">exponent</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 4.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_special"><span class="id" title="lemma">injm_special</span></a> <span class="id" title="var">D</span> <span class="id" title="var">G</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="injm_extraspecial"><span class="id" title="lemma">injm_extraspecial</span></a> <span class="id" title="var">D</span> <span class="id" title="var">G</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#efe2275bee4a5227161b40da886719a5"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#3a01b501aff42699ca141d6279e9102f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#70b0a61e30f130888503421fd44e1802"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_special"><span class="id" title="lemma">isog_special</span></a> <span class="id" title="var">G</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#special"><span class="id" title="definition">special</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="isog_extraspecial"><span class="id" title="lemma">isog_extraspecial</span></a> <span class="id" title="var">G</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>) :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="cprod_extraspecial"><span class="id" title="lemma">cprod_extraspecial</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - Lemmas bundling Aschbacher (23.10) with (19.1), (19.2), (19.12) and (20.8) -</div> -<div class="code"> -<span class="id" title="keyword">Section</span> <a name="Extraspecial.ExtraspecialFormspace"><span class="id" title="section">ExtraspecialFormspace</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypotheses</span> (<a name="Extraspecial.ExtraspecialFormspace.pG"><span class="id" title="variable">pG</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a>) (<a name="Extraspecial.ExtraspecialFormspace.esG"><span class="id" title="variable">esG</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.ExtraspecialFormspace.p_pr"><span class="id" title="variable">p_pr</span></a> := <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial_prime"><span class="id" title="lemma">extraspecial_prime</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.pG"><span class="id" title="variable">pG</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.esG"><span class="id" title="variable">esG</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.ExtraspecialFormspace.oZ"><span class="id" title="variable">oZ</span></a> := <a class="idref" href="mathcomp.solvable.maximal.html#card_center_extraspecial"><span class="id" title="lemma">card_center_extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.pG"><span class="id" title="variable">pG</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.esG"><span class="id" title="variable">esG</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.ExtraspecialFormspace.p_gt1"><span class="id" title="variable">p_gt1</span></a> := <a class="idref" href="mathcomp.ssreflect.prime.html#prime_gt1"><span class="id" title="lemma">prime_gt1</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.p_pr"><span class="id" title="variable">p_pr</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.ExtraspecialFormspace.p_gt0"><span class="id" title="variable">p_gt0</span></a> := <a class="idref" href="mathcomp.ssreflect.prime.html#prime_gt0"><span class="id" title="lemma">prime_gt0</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.p_pr"><span class="id" title="variable">p_pr</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This encasulates Aschbacher (23.10)(1). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="cent1_extraspecial_maximal"><span class="id" title="lemma">cent1_extraspecial_maximal</span></a> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#c1ad6bcc76a6221225111f87bc3b0c3d"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is the tranposition of the hyperplane dimension theorem (Aschbacher - (19.1)) to subgroups of an extraspecial group. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="subcent1_extraspecial_maximal"><span class="id" title="lemma">subcent1_extraspecial_maximal</span></a> <span class="id" title="var">U</span> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#160b1f5d5c3e3c6d274924649d717d8d"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">C_U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#bb1f7e576b21e943bdeacc0f6a28b245"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is the tranposition of the orthogonal subspace dimension theorem - (Aschbacher (19.2)) to subgroups of an extraspecial group. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="card_subcent_extraspecial"><span class="id" title="lemma">card_subcent_extraspecial</span></a> <span class="id" title="var">U</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#ea2ff3d561159081cea6fb2e8113cc54"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#0665f11b64f1431f9d664aba3c000866"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span>.<br/> - -<br/> -</div> - -<div class="doc"> - This is the tranposition of the proof that a singular vector is contained - in a hyperbolic plane (Aschbacher (19.12)) to subgroups of an extraspecial - group. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="split1_extraspecial"><span class="id" title="lemma">split1_extraspecial</span></a> <span class="id" title="var">x</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#160b1f5d5c3e3c6d274924649d717d8d"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">{</span></a><span class="id" title="var">E</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 3)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#2242f6721707980eca939ec29164eab3"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">,</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f0bbce9238fab3dd03626439080979a9"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssreflect.html#00a1a5b58aac8f1e3f1abff064a39f9d"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b76985613d3b3ca583bb3217c76109bb"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#cc5e56ba3765e2d6b17e66d19b966f1d"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is the tranposition of the proof that the dimension of any maximal - totally singular subspace is equal to the Witt index (Aschbacher (20.8)), - to subgroups of an extraspecial group (in a slightly more general form, - since we allow for p != 2). - Note that Aschbacher derives this from the Witt lemma, which we avoid. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="pmaxElem_extraspecial"><span class="id" title="lemma">pmaxElem_extraspecial</span></a> : <a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">E</span></a><a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">_p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.abelian.html#d6b709c1b89a3c62e054c9b8a6f094b4"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#7d0895628380660804762fc0f207680e"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#7d0895628380660804762fc0f207680e"><span class="id" title="notation">E_p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#7d0895628380660804762fc0f207680e"><span class="id" title="notation">^(</span></a><a class="idref" href="mathcomp.solvable.abelian.html#6b61dcfb093dfe93d87341f88d96ca9f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#6b61dcfb093dfe93d87341f88d96ca9f"><span class="id" title="notation">r_p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#6b61dcfb093dfe93d87341f88d96ca9f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.abelian.html#6b61dcfb093dfe93d87341f88d96ca9f"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.abelian.html#7d0895628380660804762fc0f207680e"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.abelian.html#7d0895628380660804762fc0f207680e"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.ExtraspecialFormspace"><span class="id" title="section">ExtraspecialFormspace</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is B & G, Theorem 4.15, as done in Aschbacher (23.8) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="critical_extraspecial"><span class="id" title="lemma">critical_extraspecial</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#653e048e978e57b1e513b9d5de2caee6"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#R"><span class="id" title="variable">R</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is part of Aschbacher (23.13) and (23.14). -</div> -<div class="code"> -<span class="id" title="keyword">Theorem</span> <a name="extraspecial_structure"><span class="id" title="lemma">extraspecial_structure</span></a> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">{</span></a><span class="id" title="var">Es</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">E</span> ⇒ (<a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> 3)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#df45e8c2e8370fd4f0f7c4fdaf208180"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#9ddeac0ab66152bd1d64bedb507a795e"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.solvable.maximal.html#Es"><span class="id" title="variable">Es</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">/</span></a>1%<span class="id" title="var">g</span><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">(</span></a><span class="id" title="var">E</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Es"><span class="id" title="variable">Es</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#93a42d9430a115f2544a09cba4cf05ca"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#1c2e0971edf6e9b6c6dd4a5951d04f36"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="Extraspecial.StructureCorollaries"><span class="id" title="section">StructureCorollaries</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variable</span> <a name="Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypotheses</span> (<a name="Extraspecial.StructureCorollaries.pS"><span class="id" title="variable">pS</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a>) (<a name="Extraspecial.StructureCorollaries.esS"><span class="id" title="variable">esS</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial"><span class="id" title="definition">extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.StructureCorollaries.p_pr"><span class="id" title="variable">p_pr</span></a> := <a class="idref" href="mathcomp.solvable.maximal.html#extraspecial_prime"><span class="id" title="lemma">extraspecial_prime</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.pS"><span class="id" title="variable">pS</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.esS"><span class="id" title="variable">esS</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="Extraspecial.StructureCorollaries.oZ"><span class="id" title="variable">oZ</span></a> := <a class="idref" href="mathcomp.solvable.maximal.html#card_center_extraspecial"><span class="id" title="lemma">card_center_extraspecial</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.pS"><span class="id" title="variable">pS</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.esS"><span class="id" title="variable">esS</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher (23.10)(2). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="card_extraspecial"><span class="id" title="lemma">card_extraspecial</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">{</span></a><span class="id" title="var">n</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#7f2a7ef2c63af7359b22787a9daf336e"><span class="id" title="notation">></span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#234f50e13366f794cd6877cf832a5935"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#81fd94e251a61ee523cdd7855774ae7c"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bff172cdafaf4b86cefb300b16285e42"><span class="id" title="notation">.*2</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#bda89d73ec4a8f23ae92b565ffb5aaa6"><span class="id" title="notation">.+1</span></a>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#c0bbd202248f4def7aaf0c316cf2c29e"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="Aut_extraspecial_full"><span class="id" title="lemma">Aut_extraspecial_full</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#Aut_in"><span class="id" title="definition">Aut_in</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a>) <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#13d63916ddaa339df3fcf04363ae7cde"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - These are the parts of Aschbacher (23.12) and exercise 8.5 that are later - used in Aschbacher (34.9), which itself replaces the informal discussion - quoted from Gorenstein in the proof of B & G, Theorem 2.5. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="center_aut_extraspecial"><span class="id" title="lemma">center_aut_extraspecial</span></a> <span class="id" title="var">k</span> : <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a><br/> - <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#3df228c109f14f0423b4fccc967ee1ac"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">z</span>, <a class="idref" href="mathcomp.solvable.maximal.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries.S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.solvable.center.html#e90cc03a62af307fc4e121114703663b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#06cdd2633d7788bac7abeac13b2dd91e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">g</span>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial.StructureCorollaries"><span class="id" title="section">StructureCorollaries</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#Extraspecial"><span class="id" title="section">Extraspecial</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Section</span> <a name="SCN"><span class="id" title="section">SCN</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> (<a name="SCN.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="SCN.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a name="SCN.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>).<br/> -<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">Z</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="SCN_P"><span class="id" title="lemma">SCN_P</span></a> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">SCN</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">)</span></a>).<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="SCN_abelian"><span class="id" title="lemma">SCN_abelian</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">SCN</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="exponent_Ohm1_class2"><span class="id" title="lemma">exponent_Ohm1_class2</span></a> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#exponent"><span class="id" title="definition">exponent</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">Ohm_1</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#bde82eab2fe4a0799bc2419e587505d4"><span class="id" title="notation">%|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - SCN_max and max_SCN cover Aschbacher 23.15(1) -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="SCN_max"><span class="id" title="lemma">SCN_max</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">SCN</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">max</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">]</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="max_SCN"><span class="id" title="lemma">max_SCN</span></a> <span class="id" title="var">A</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">max</span></a> <span class="id" title="var">A</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">SCN</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - The two other assertions of Aschbacher 23.15 state properties of the - normal series 1 <| Z = 'Ohm_1(A) <| A with A \in 'SCN(G). -</div> -<div class="code"> - -<br/> -<span class="id" title="keyword">Section</span> <a name="SCN.SCNseries"><span class="id" title="section">SCNseries</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Variables</span> <a name="SCN.SCNseries.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a>.<br/> -<span class="id" title="keyword">Hypothesis</span> <a name="SCN.SCNseries.SCN_A"><span class="id" title="variable">SCN_A</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.ssr.ssrbool.html#b09457274bcb94927e289b8a9e9cd3f7"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">SCN</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.maximal.html#d89bb215e1dc40c58e36a9bc7fe6a2ce"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Let</span> <a name="SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a> := <a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">Ohm_1</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">)</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="SCN.SCNseries.cAA"><span class="id" title="variable">cAA</span></a> := <a class="idref" href="mathcomp.solvable.maximal.html#SCN_abelian"><span class="id" title="lemma">SCN_abelian</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.SCN_A"><span class="id" title="variable">SCN_A</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="SCN.SCNseries.sZA"><span class="id" title="variable">sZA</span></a>: <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a> := <a class="idref" href="mathcomp.solvable.abelian.html#Ohm_sub"><span class="id" title="lemma">Ohm_sub</span></a> 1 <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a>.<br/> -<span class="id" title="keyword">Let</span> <a name="SCN.SCNseries.nZA"><span class="id" title="variable">nZA</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1ff9e060a8cc6098d64e42214fa57c96"><span class="id" title="notation">)</span></a> := <a class="idref" href="mathcomp.fingroup.fingroup.html#sub_abelian_norm"><span class="id" title="lemma">sub_abelian_norm</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.cAA"><span class="id" title="variable">cAA</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.sZA"><span class="id" title="variable">sZA</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher 23.15(2). -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="der1_stab_Ohm1_SCN_series"><span class="id" title="lemma">der1_stab_Ohm1_SCN_series</span></a> : <a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#313ef60ac6c7566906fa5b28c1bbf405"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b9596739b058766532fc6517a36fef9f"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f8cc9e9a87e2166483438ca61952b8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f8cc9e9a87e2166483438ca61952b8a4"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)^`(</span></a>1<a class="idref" href="mathcomp.solvable.commutator.html#5684e4e024467813e860f228f2381620"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher 23.15(3); note that this proof does not depend on the - maximality of A. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Ohm1_stab_Ohm1_SCN_series"><span class="id" title="lemma">Ohm1_stab_Ohm1_SCN_series</span></a> :<br/> - <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">Ohm_1</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#4102da6205bd8605932488256a8bd517"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#3e65ad3edf5f7fb3ea6bc63a878112a8"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries.Z"><span class="id" title="variable">Z</span></a> <a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f8cc9e9a87e2166483438ca61952b8a4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f8cc9e9a87e2166483438ca61952b8a4"><span class="id" title="notation">Q</span></a><a class="idref" href="mathcomp.fingroup.action.html#6998185c5f7a5efd284e9c25133c5e71"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.SCNseries"><span class="id" title="section">SCNseries</span></a>.<br/> - -<br/> -</div> - -<div class="doc"> - This is Aschbacher 23.16. -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Ohm1_cent_max_normal_abelem"><span class="id" title="lemma">Ohm1_cent_max_normal_abelem</span></a> <span class="id" title="var">Z</span> :<br/> - <a class="idref" href="mathcomp.ssreflect.ssrnat.html#odd"><span class="id" title="definition">odd</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">max</span></a> <span class="id" title="var">Z</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Z"><span class="id" title="variable">Z</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7e8095b432e7aa5c3c22bb87584658b7"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#9926250b7ba3fd427de487631b06d875"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Z"><span class="id" title="variable">Z</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab4e6ccf257de3bf5b59570ca8a7e280"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">Ohm_1</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#Z"><span class="id" title="variable">Z</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#04a5555c0db8685a27679a7e6af3f8c3"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.solvable.abelian.html#c56ec4cf607c781766b0d2cf7a260ba8"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#Z"><span class="id" title="variable">Z</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="critical_class2"><span class="id" title="lemma">critical_class2</span></a> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.maximal.html#critical"><span class="id" title="definition">critical</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.nilpotent.html#nil_class"><span class="id" title="definition">nil_class</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#cb53cf0ee22c036a03b4a9281c68b5a3"><span class="id" title="notation">≤</span></a> 2.<br/> - -<br/> -</div> - -<div class="doc"> - This proof of the Thompson critical lemma is adapted from Aschbacher 23.6 -</div> -<div class="code"> -<span class="id" title="keyword">Lemma</span> <a name="Thompson_critical"><span class="id" title="lemma">Thompson_critical</span></a> : <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">{</span></a><span class="id" title="var">K</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#dd8cd2228f051940101d045bfdffe2d9"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#critical"><span class="id" title="definition">critical</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Specif.html#6556914db359db999889decec6a4a562"><span class="id" title="notation">}</span></a>.<br/> - -<br/> -<span class="id" title="keyword">Lemma</span> <a name="critical_p_stab_Aut"><span class="id" title="lemma">critical_p_stab_Aut</span></a> <span class="id" title="var">H</span> :<br/> - <a class="idref" href="mathcomp.solvable.maximal.html#critical"><span class="id" title="definition">critical</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.9.0/stdlib//Coq.Init.Logic.html#1c93e43e07fbeaeb6a625cb6614beb5d"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#15605b2ce8a0bd336aafa96c5cc1afdc"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.action.html#563ec7f167b9e19c804c7f8a07d81e1d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#563ec7f167b9e19c804c7f8a07d81e1d"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#563ec7f167b9e19c804c7f8a07d81e1d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.maximal.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.action.html#563ec7f167b9e19c804c7f8a07d81e1d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.action.html#c6d4b1f79e75b84124452166fdbe2f47"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#c6d4b1f79e75b84124452166fdbe2f47"><span class="id" title="notation">Aut</span></a> <a class="idref" href="mathcomp.solvable.maximal.html#SCN.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.action.html#c6d4b1f79e75b84124452166fdbe2f47"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.fingroup.action.html#563ec7f167b9e19c804c7f8a07d81e1d"><span class="id" title="notation">)</span></a>.<br/> - -<br/> -<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.maximal.html#SCN"><span class="id" title="section">SCN</span></a>.<br/> - -<br/> -</div> -</div> - -<div id="footer"> -<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> -</div> - -</div> - -</body> -</html>
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