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| author | Enrico Tassi | 2018-04-20 10:54:22 +0200 |
|---|---|---|
| committer | Enrico Tassi | 2018-04-20 10:54:22 +0200 |
| commit | ed05182cece6bb3706e09b2ce14af4a41a2e8141 (patch) | |
| tree | e850d7314b6372d0476cf2ffaf7d3830721db7b1 /docs/htmldoc/mathcomp.fingroup.automorphism.html | |
| parent | 3d196f44681fb3b23ff8a79fbd44e12308680531 (diff) | |
generate the documentation for 1.7
Diffstat (limited to 'docs/htmldoc/mathcomp.fingroup.automorphism.html')
| -rw-r--r-- | docs/htmldoc/mathcomp.fingroup.automorphism.html | 491 |
1 files changed, 491 insertions, 0 deletions
diff --git a/docs/htmldoc/mathcomp.fingroup.automorphism.html b/docs/htmldoc/mathcomp.fingroup.automorphism.html new file mode 100644 index 0000000..bff22e2 --- /dev/null +++ b/docs/htmldoc/mathcomp.fingroup.automorphism.html @@ -0,0 +1,491 @@ +<!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.fingroup.automorphism</title> +</head> + +<body> + +<div id="page"> + +<div id="header"> +</div> + +<div id="main"> + +<h1 class="libtitle">Library mathcomp.fingroup.automorphism</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/> +<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Group automorphisms and characteristic subgroups. + Unlike morphisms on a group G, which are functions of type gT -> rT, with + a canonical structure of dependent type {morphim G >-> rT}, automorphisms + are permutations of type {perm gT} contained in Aut G : {set {perm gT}}. + This lets us use the finGroupType of {perm gT}. Note also that while + morphisms on G are undefined outside G, automorphisms have their support + in G, i.e., they are the identity outside G. + Definitions: + Aut G (or [Aut G]) == the automorphism group of G. + [Aut G]%G == the group structure for Aut G. + autm AutGa == the morphism on G induced by a, given + AutGa : a \in Aut G. + perm_in injf fA == the permutation with support B in induced by f, + given injf : {in A &, injective f} and + fA : f @: A \subset A. + aut injf fG == the automorphism of G induced by the morphism f, + given injf : 'injm f and fG : f @* G \subset G. + Aut_isom injf sDom == the injective homomorphism that maps Aut G to + Aut (f @* G), with f : {morphism D >-> rT} and + given injf: 'injm f and sDom : G \subset D. + conjgm G == the conjugation automorphism on G. + H \char G == H is a characteristic subgroup of G. +</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/> +</div> + +<div class="doc"> + A group automorphism, defined as a permutation on a subset of a + finGroupType that respects the morphism law. + Here perm_on is used as a closure rule for the set A. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Automorphism"><span class="id" title="section">Automorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Automorphism.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">A</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><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">b</span> : <a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">perm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Aut"><span class="id" title="definition">Aut</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.ssreflect.finset.html#d8251b84d533463822c9c970f6c348cb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8251b84d533463822c9c970f6c348cb"><span class="id" title="notation">set</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#d8251b84d533463822c9c970f6c348cb"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.perm.html#perm_on"><span class="id" title="definition">perm_on</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#d8251b84d533463822c9c970f6c348cb"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#d8251b84d533463822c9c970f6c348cb"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_morphic"><span class="id" title="lemma">Aut_morphic</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="out_Aut"><span class="id" title="lemma">out_Aut</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ad6d23746eb1a3b62e52010d3945a1db"><span class="id" title="notation">notin</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="eq_Aut"><span class="id" title="lemma">eq_Aut</span></a> <span class="id" title="var">A</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><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.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + The morphism that is represented by a given element of Aut A. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="autm"><span class="id" title="definition">autm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> (<span class="id" title="var">AutAa</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>) := <a class="idref" href="mathcomp.fingroup.morphism.html#morphm"><span class="id" title="definition">morphm</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_morphic"><span class="id" title="lemma">Aut_morphic</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutAa"><span class="id" title="variable">AutAa</span></a>).<br/> +<span class="id" title="keyword">Lemma</span> <a name="autmE"><span class="id" title="lemma">autmE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">a</span> (<span class="id" title="var">AutAa</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="mathcomp.fingroup.automorphism.html#autm"><span class="id" title="definition">autm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutAa"><span class="id" title="variable">AutAa</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">autm_morphism</span> <span class="id" title="var">A</span> <span class="id" title="var">a</span> <span class="id" title="var">aM</span> := <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.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> @<a class="idref" href="mathcomp.fingroup.automorphism.html#autm"><span class="id" title="definition">autm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aM"><span class="id" title="variable">aM</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Automorphism.AutGroup"><span class="id" title="section">AutGroup</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_group_set"><span class="id" title="lemma">Aut_group_set</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Aut_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_group_set"><span class="id" title="lemma">Aut_group_set</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> (<a name="Automorphism.AutGroup.a"><span class="id" title="variable">a</span></a> : <a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">perm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.perm.html#8787ba8c1ae558dde17e1bdb8427679e"><span class="id" title="notation">}</span></a>) (<a name="Automorphism.AutGroup.AutGa"><span class="id" title="variable">AutGa</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="f"><span class="id" title="abbreviation">f</span></a> := (<a class="idref" href="mathcomp.fingroup.automorphism.html#autm"><span class="id" title="definition">autm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.AutGa"><span class="id" title="variable">AutGa</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="fE"><span class="id" title="abbreviation">fE</span></a> := (<a class="idref" href="mathcomp.fingroup.automorphism.html#autmE"><span class="id" title="lemma">autmE</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.AutGa"><span class="id" title="variable">AutGa</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_autm"><span class="id" title="lemma">injm_autm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="abbreviation">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_autm"><span class="id" title="lemma">ker_autm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_autm"><span class="id" title="lemma">im_autm</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_closed"><span class="id" title="lemma">Aut_closed</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism.AutGroup"><span class="id" title="section">AutGroup</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut1"><span class="id" title="lemma">Aut1</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#Automorphism"><span class="id" title="section">Automorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="3c9674036656e61bc8c8b776f748d7f1"><span class="id" title="notation">"</span></a>[ 'Aut' G ]" := (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_group"><span class="id" title="definition">Aut_group</span></a> <span class="id" title="var">G</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'Aut' G ]") : <span class="id" title="var">Group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="3da384943d68ab3a8086ce37ca6f27fc"><span class="id" title="notation">"</span></a>[ 'Aut' G ]" := (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <span class="id" title="var">G</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">only</span> <span class="id" title="var">parsing</span>) : <span class="id" title="var">group_scope</span>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + The permutation function (total on the underlying groupType) that is the + representant of a given morphism f with domain A in (Aut A). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="PermIn"><span class="id" title="section">PermIn</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="PermIn.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<a name="PermIn.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="PermIn.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#T"><span class="id" title="variable">T</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypotheses</span> (<a name="PermIn.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>) (<a name="PermIn.sBf"><span class="id" title="variable">sBf</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="perm_in_inj"><span class="id" title="lemma">perm_in_inj</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="perm_in"><span class="id" title="definition">perm_in</span></a> := <a class="idref" href="mathcomp.fingroup.perm.html#perm"><span class="id" title="abbreviation">perm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#perm_in_inj"><span class="id" title="lemma">perm_in_inj</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="perm_in_on"><span class="id" title="lemma">perm_in_on</span></a> : <a class="idref" href="mathcomp.fingroup.perm.html#perm_on"><span class="id" title="definition">perm_on</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#perm_in"><span class="id" title="definition">perm_in</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="perm_inE"><span class="id" title="lemma">perm_inE</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#perm_in"><span class="id" title="definition">perm_in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#PermIn"><span class="id" title="section">PermIn</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + properties of injective endomorphisms +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MakeAut"><span class="id" title="section">MakeAut</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="MakeAut.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="MakeAut.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MakeAut.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="MakeAut.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_fixP"><span class="id" title="lemma">morphim_fixP</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="MakeAut.Gf"><span class="id" title="variable">Gf</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="aut_closed"><span class="id" title="lemma">aut_closed</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="aut"><span class="id" title="definition">aut</span></a> := <a class="idref" href="mathcomp.fingroup.automorphism.html#perm_in"><span class="id" title="definition">perm_in</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#injmP"><span class="id" title="lemma">injmP</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.injf"><span class="id" title="variable">injf</span></a>) <a class="idref" href="mathcomp.fingroup.automorphism.html#aut_closed"><span class="id" title="lemma">aut_closed</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="autE"><span class="id" title="lemma">autE</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphic_aut"><span class="id" title="lemma">morphic_aut</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_aut"><span class="id" title="lemma">Aut_aut</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="imset_autE"><span class="id" title="lemma">imset_autE</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="preim_autE"><span class="id" title="lemma">preim_autE</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#MakeAut"><span class="id" title="section">MakeAut</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="AutIsom"><span class="id" title="section">AutIsom</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="AutIsom.gT"><span class="id" title="variable">gT</span></a> <a name="AutIsom.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="AutIsom.G"><span class="id" title="variable">G</span></a> <a name="AutIsom.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="AutIsom.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypotheses</span> (<a name="AutIsom.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a>) (<a name="AutIsom.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.D"><span class="id" title="variable">D</span></a>).<br/> +<span class="id" title="keyword">Let</span> <a name="AutIsom.domG"><span class="id" title="variable">domG</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#subsetP"><span class="id" title="lemma">subsetP</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.sGD"><span class="id" title="variable">sGD</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_isom_subproof"><span class="id" title="lemma">Aut_isom_subproof</span></a> <span class="id" title="var">a</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">a'</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a'"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a'"><span class="id" title="variable">a'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="Aut_isom"><span class="id" title="definition">Aut_isom</span></a> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#s2val"><span class="id" title="definition">s2val</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom_subproof"><span class="id" title="lemma">Aut_isom_subproof</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_Aut_isom"><span class="id" title="lemma">Aut_Aut_isom</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_isomE"><span class="id" title="lemma">Aut_isomE</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><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.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_isomM"><span class="id" title="lemma">Aut_isomM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><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.fingroup.automorphism.html#AutIsom.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Aut_isom_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isomM"><span class="id" title="lemma">Aut_isomM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_Aut_isom"><span class="id" title="lemma">injm_Aut_isom</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#AutIsom"><span class="id" title="section">AutIsom</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InjmAut"><span class="id" title="section">InjmAut</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="InjmAut.gT"><span class="id" title="variable">gT</span></a> <a name="InjmAut.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="InjmAut.G"><span class="id" title="variable">G</span></a> <a name="InjmAut.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="InjmAut.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypotheses</span> (<a name="InjmAut.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.f"><span class="id" title="variable">f</span></a>) (<a name="InjmAut.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.D"><span class="id" title="variable">D</span></a>).<br/> +<span class="id" title="keyword">Let</span> <a name="InjmAut.domG"><span class="id" title="variable">domG</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#subsetP"><span class="id" title="lemma">subsetP</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.sGD"><span class="id" title="variable">sGD</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_Aut_isom"><span class="id" title="lemma">im_Aut_isom</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.injf"><span class="id" title="variable">injf</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.sGD"><span class="id" title="variable">sGD</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_isomP"><span class="id" title="lemma">Aut_isomP</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a>)) (<a class="idref" href="mathcomp.fingroup.automorphism.html#Aut_isom"><span class="id" title="definition">Aut_isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.injf"><span class="id" title="variable">injf</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.sGD"><span class="id" title="variable">sGD</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_Aut"><span class="id" title="lemma">injm_Aut</span></a> : <a class="idref" href="mathcomp.fingroup.automorphism.html#Aut"><span class="id" title="definition">Aut</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><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.fingroup.automorphism.html#InjmAut.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmAut"><span class="id" title="section">InjmAut</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + conjugation automorphism +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ConjugationMorphism"><span class="id" title="section">ConjugationMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ConjugationMorphism.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">A</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="conjgm"><span class="id" title="definition">conjgm</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.gT"><span class="id" title="variable">gT</span></a> ⇒ <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="conjgmE"><span class="id" title="lemma">conjgmE</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">conjgm_morphism</span> <span class="id" title="var">A</span> <span class="id" title="var">x</span> :=<br/> + @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span> ⇒ <a class="idref" href="mathcomp.fingroup.fingroup.html#conjMg"><span class="id" title="lemma">conjMg</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_conj"><span class="id" title="lemma">morphim_conj</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ConjugationMorphism.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_conj"><span class="id" title="lemma">injm_conj</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="conj_isom"><span class="id" title="lemma">conj_isom</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="conj_isog"><span class="id" title="lemma">conj_isog</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="norm_conjg_im"><span class="id" title="lemma">norm_conjg_im</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="norm_conj_isom"><span class="id" title="lemma">norm_conj_isom</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#conjgm"><span class="id" title="definition">conjgm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="conj_aut"><span class="id" title="definition">conj_aut</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.fingroup.automorphism.html#aut"><span class="id" title="definition">aut</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#injm_conj"><span class="id" title="lemma">injm_conj</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="mathcomp.fingroup.automorphism.html#norm_conjg_im"><span class="id" title="lemma">norm_conjg_im</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#subgP"><span class="id" title="lemma">subgP</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a>))).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="norm_conj_autE"><span class="id" title="lemma">norm_conj_autE</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut"><span class="id" title="definition">conj_aut</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="conj_autE"><span class="id" title="lemma">conj_autE</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut"><span class="id" title="definition">conj_aut</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="conj_aut_morphM"><span class="id" title="lemma">conj_aut_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut"><span class="id" title="definition">conj_aut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">conj_aut_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut_morphM"><span class="id" title="lemma">conj_aut_morphM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_conj_aut"><span class="id" title="lemma">ker_conj_aut</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut"><span class="id" title="definition">conj_aut</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="Aut_conj_aut"><span class="id" title="lemma">Aut_conj_aut</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#conj_aut"><span class="id" title="definition">conj_aut</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><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.fingroup.automorphism.html#ConjugationMorphism.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#ConjugationMorphism"><span class="id" title="section">ConjugationMorphism</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Reserved Notation</span> "G \char H" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<br/> + +<br/> +</div> + +<div class="doc"> + Characteristic subgroup +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Characteristicity"><span class="id" title="section">Characteristicity</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="Characteristicity.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> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Characteristicity.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><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">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#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Characteristicity.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="characteristic"><span class="id" title="definition">characteristic</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/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">∀</span></a> <span class="id" title="var">f</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><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.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Infix</span> <a name="fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">"</span></a>\char" := <a class="idref" href="mathcomp.fingroup.automorphism.html#characteristic"><span class="id" title="definition">characteristic</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="charP"><span class="id" title="lemma">charP</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/> + <span class="id" title="keyword">let</span> <span class="id" title="var">fixH</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#Characteristicity.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) := <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <span class="id" title="tactic">in</span><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">f</span>, <a class="idref" href="mathcomp.fingroup.automorphism.html#fixH"><span class="id" title="variable">fixH</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4d70beb460e783df2b7b2ad387c00194"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Characteristic subgroup properties : composition, relational properties +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char1"><span class="id" title="lemma">char1</span></a> <span class="id" title="var">G</span> : 1 <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_refl"><span class="id" title="lemma">char_refl</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_trans"><span class="id" title="lemma">char_trans</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_norms"><span class="id" title="lemma">char_norms</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_sub"><span class="id" title="lemma">char_sub</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_norm_trans"><span class="id" title="lemma">char_norm_trans</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_normal_trans"><span class="id" title="lemma">char_normal_trans</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_normal"><span class="id" title="lemma">char_normal</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_norm"><span class="id" title="lemma">char_norm</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="charI"><span class="id" title="lemma">charI</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="charY"><span class="id" title="lemma">charY</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="charM"><span class="id" title="lemma">charM</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="lone_subgroup_char"><span class="id" title="lemma">lone_subgroup_char</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><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.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#fc41e85c800db79ed681728de9fb8a9b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#Characteristicity"><span class="id" title="section">Characteristicity</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">"</span></a>H \char G" := (<a class="idref" href="mathcomp.fingroup.automorphism.html#characteristic"><span class="id" title="definition">characteristic</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span>) : <span class="id" title="var">group_scope</span>.<br/> +<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">char_refl</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InjmChar"><span class="id" title="section">InjmChar</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="InjmChar.aT"><span class="id" title="variable">aT</span></a> <a name="InjmChar.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="InjmChar.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="InjmChar.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="InjmChar.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_char"><span class="id" title="lemma">injm_char</span></a> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#InjmChar"><span class="id" title="section">InjmChar</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="CharInjm"><span class="id" title="section">CharInjm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="CharInjm.aT"><span class="id" title="variable">aT</span></a> <a name="CharInjm.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="CharInjm.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="CharInjm.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="CharInjm.injf"><span class="id" title="variable">injf</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="char_injm"><span class="id" title="lemma">char_injm</span></a> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.automorphism.html#004858100bfba9714bde1cdbce60358b"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.fingroup.automorphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.automorphism.html#CharInjm"><span class="id" title="section">CharInjm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Unset Implicit Arguments</span>.<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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