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<h1 class="libtitle">Library mathcomp.solvable.gseries</h1>

<div class="code">
<span class="comment">(*&nbsp;(c)&nbsp;Copyright&nbsp;2006-2016&nbsp;Microsoft&nbsp;Corporation&nbsp;and&nbsp;Inria.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
&nbsp;Distributed&nbsp;under&nbsp;the&nbsp;terms&nbsp;of&nbsp;CeCILL-B.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*)</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/>

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</div>

<div class="doc">
           H &lt;|&lt;| G   &lt;=&gt; H is subnormal in G, i.e., H &lt;| ... &lt;| G.         
 invariant_factor A H G &lt;=&gt; A normalises both H and G, and H &lt;| G.          
         A.-invariant &lt;=&gt; the (invariant_factor A) relation, in the context 
                          of the g_rel.-series notation.                    
    g_rel.-series H s &lt;=&gt; H :: s is a sequence of groups whose projection   
                          to sets satisfies relation g_rel pairwise; for    
                          example H &lt;|&lt;| G iff G = last H s for some s such 
                          that normal.-series H s.                          
   stable_factor A H G == H &lt;| G and A centralises G / H.                   
             A.-stable == the stable_factor relation, in the scope of the   
                          r.-series notation.                               
            G.-central == the central_factor relation, in the scope of the  
                          r.-series notation.                               
           maximal M G == M is a maximal proper subgroup of G.              
        maximal_eq M G == (M == G) or (maximal M G).                        
       maxnormal M G N == M is a maximal subgroup of G normalized by N.     
         minnormal M N == M is a minimal nontrivial group normalized by N.  
              simple G == G is a (nontrivial) simple group.                 
                       := minnormal G G                                     
              G.-chief == the chief_factor relation, in the scope of the    
                          r.-series notation.                                
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>

<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="GroupDefs"><span class="id" title="section">GroupDefs</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="GroupDefs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">U</span> <span class="id" title="var">V</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.solvable.gseries.html#GroupDefs.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/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="subnormal"><span class="id" title="definition">subnormal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/>
&nbsp;&nbsp;<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.solvable.gseries.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.solvable.gseries.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">&amp;&amp;</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.ssrnat.html#iter"><span class="id" title="definition">iter</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">N</span> ⇒ <a class="idref" href="mathcomp.fingroup.fingroup.html#generated"><span class="id" title="definition">generated</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#class_support"><span class="id" title="definition">class_support</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#N"><span class="id" title="variable">N</span></a>)) <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="invariant_factor"><span class="id" title="definition">invariant_factor</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#B"><span class="id" title="variable">B</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#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#C"><span class="id" title="variable">C</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#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="group_rel_of"><span class="id" title="definition">group_rel_of</span></a> (<span class="id" title="var">r</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel"><span class="id" title="definition">rel</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.solvable.gseries.html#GroupDefs.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>) := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">rel</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#groupT"><span class="id" title="abbreviation">groupT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#a3fa885266d530b93c1b9f67891df8e8"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="stable_factor"><span class="id" title="definition">stable_factor</span></a> <span class="id" title="var">A</span> <span class="id" title="var">V</span> <span class="id" title="var">U</span> :=<br/>
&nbsp;&nbsp;<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.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><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.solvable.gseries.html#V"><span class="id" title="variable">V</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">&amp;&amp;</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.solvable.gseries.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</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>. <span class="comment">(*&nbsp;this&nbsp;orders&nbsp;allows&nbsp;and3P&nbsp;to&nbsp;be&nbsp;used&nbsp;*)</span><br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="central_factor"><span class="id" title="definition">central_factor</span></a> <span class="id" title="var">A</span> <span class="id" title="var">V</span> <span class="id" title="var">U</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&amp;&amp;</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><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.solvable.gseries.html#V"><span class="id" title="variable">V</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</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.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</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.solvable.gseries.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#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="maximal"><span class="id" title="definition">maximal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">max</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">of</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#c203c753b6e0cf91b69de08395409b1a"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <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.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#14a7a9c7dc61f86bfb664d400fabaf8a"><span class="id" title="notation">||</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="maxnormal"><span class="id" title="definition">maxnormal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">U</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">max</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">of</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</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.solvable.gseries.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.fingroup.fingroup.html#606edfc5376586561212e0b1e908a07b"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="minnormal"><span class="id" title="definition">minnormal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">min</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">of</span></a> <span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="mathcomp.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</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.solvable.gseries.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.fingroup.fingroup.html#ff34d14fa7e7764d47d58a9547aa60ae"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="simple"><span class="id" title="definition">simple</span></a> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="chief_factor"><span class="id" title="definition">chief_factor</span></a> <span class="id" title="var">A</span> <span class="id" title="var">V</span> <span class="id" title="var">U</span> := <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</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.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#GroupDefs"><span class="id" title="section">GroupDefs</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&quot;</span></a>H &lt;|&lt;| G" := (<a class="idref" href="mathcomp.solvable.gseries.html#subnormal"><span class="id" title="definition">subnormal</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">group_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="2929694f01eaa280a75b30feb6fcabad"><span class="id" title="notation">&quot;</span></a>A .-invariant" := (<a class="idref" href="mathcomp.solvable.gseries.html#invariant_factor"><span class="id" title="definition">invariant_factor</span></a> <span class="id" title="var">A</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "A .-invariant") : <span class="id" title="var">group_rel_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="fa2b3265fe9713983c491442558795f1"><span class="id" title="notation">&quot;</span></a>A .-stable" := (<a class="idref" href="mathcomp.solvable.gseries.html#stable_factor"><span class="id" title="definition">stable_factor</span></a> <span class="id" title="var">A</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "A .-stable") : <span class="id" title="var">group_rel_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="d5eabd0e3d55ff510a60634b70f7664c"><span class="id" title="notation">&quot;</span></a>A .-central" := (<a class="idref" href="mathcomp.solvable.gseries.html#central_factor"><span class="id" title="definition">central_factor</span></a> <span class="id" title="var">A</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "A .-central") : <span class="id" title="var">group_rel_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="34031cd5d91f4c8b27b08f87cc5b5aaf"><span class="id" title="notation">&quot;</span></a>G .-chief" := (<a class="idref" href="mathcomp.solvable.gseries.html#chief_factor"><span class="id" title="definition">chief_factor</span></a> <span class="id" title="var">G</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "G .-chief") : <span class="id" title="var">group_rel_scope</span>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">&quot;</span></a>r .-series" := (<a class="idref" href="mathcomp.ssreflect.path.html#path"><span class="id" title="definition">path</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#rel_of_simpl_rel"><span class="id" title="definition">rel_of_simpl_rel</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#group_rel_of"><span class="id" title="definition">group_rel_of</span></a> <span class="id" title="var">r</span>)))<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "r .-series") : <span class="id" title="var">group_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Subnormal"><span class="id" title="section">Subnormal</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="Subnormal.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> <span class="id" title="var">D</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.solvable.gseries.html#Subnormal.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="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</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.solvable.gseries.html#Subnormal.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">Let</span> <a name="Subnormal.setIgr"><span class="id" title="variable">setIgr</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> := (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#afecdac1581fada66d09c1ab40cfc23e"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a>)%<span class="id" title="var">G</span>.<br/>
<span class="id" title="keyword">Let</span> <a name="Subnormal.sub_setIgr"><span class="id" title="variable">sub_setIgr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.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.solvable.gseries.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.solvable.gseries.html#Subnormal.setIgr"><span class="id" title="variable">setIgr</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Let</span> <a name="Subnormal.path_setIgr"><span class="id" title="variable">path_setIgr</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> <span class="id" title="var">s</span> :<br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.fingroup.html#normal"><span class="id" title="definition">normal</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</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#normal"><span class="id" title="definition">normal</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#Subnormal.setIgr"><span class="id" title="variable">setIgr</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#Subnormal.setIgr"><span class="id" title="variable">setIgr</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>) <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormalP"><span class="id" title="lemma">subnormalP</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/>
&nbsp;&nbsp;<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.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#normal"><span class="id" title="definition">normal</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormal_refl"><span class="id" title="lemma">subnormal_refl</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormal_trans"><span class="id" title="lemma">subnormal_trans</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="normal_subnormal"><span class="id" title="lemma">normal_subnormal</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="setI_subnormal"><span class="id" title="lemma">setI_subnormal</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.solvable.gseries.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.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&amp;:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormal_sub"><span class="id" title="lemma">subnormal_sub</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="invariant_subnormal"><span class="id" title="lemma">invariant_subnormal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.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.solvable.gseries.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.solvable.gseries.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> <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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.solvable.gseries.html#2929694f01eaa280a75b30feb6fcabad"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#2929694f01eaa280a75b30feb6fcabad"><span class="id" title="notation">invariant</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">).-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormalEsupport"><span class="id" title="lemma">subnormalEsupport</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&lt;&lt;</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#class_support"><span class="id" title="definition">class_support</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">&gt;&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormalEr"><span class="id" title="lemma">subnormalEr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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> <br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">K</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.solvable.gseries.html#Subnormal.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="subnormalEl"><span class="id" title="lemma">subnormalEl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">K</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.solvable.gseries.html#Subnormal.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#Subnormal"><span class="id" title="section">Subnormal</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MorphSubNormal"><span class="id" title="section">MorphSubNormal</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="MorphSubNormal.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</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.solvable.gseries.html#MorphSubNormal.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="morphim_subnormal"><span class="id" title="lemma">morphim_subnormal</span></a> (<span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) <span class="id" title="var">G</span> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.solvable.gseries.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>) <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#K"><span class="id" title="variable">K</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="quotient_subnormal"><span class="id" title="lemma">quotient_subnormal</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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#1c13d5a750ea3c2ea3c64e0c79244ba5"><span class="id" title="notation">&lt;|&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#MorphSubNormal"><span class="id" title="section">MorphSubNormal</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MaxProps"><span class="id" title="section">MaxProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="MaxProps.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">G</span> <span class="id" title="var">H</span> <span class="id" title="var">M</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.solvable.gseries.html#MaxProps.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="maximal_eqP"><span class="id" title="lemma">maximal_eqP</span></a> <span class="id" title="var">M</span> <span class="id" title="var">G</span> :<br/>
&nbsp;&nbsp;<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.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <span class="id" title="var">H</span>, <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.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.solvable.gseries.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.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maximal_exists"><span class="id" title="lemma">maximal_exists</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.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><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">M</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">:</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.solvable.gseries.html#MaxProps.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.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#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="mulg_normal_maximal"><span class="id" title="lemma">mulg_normal_maximal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">M</span> <span class="id" title="var">H</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.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.solvable.gseries.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.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">~~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#4b80c70cdb231351c5e129ba61f7f956"><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.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>)%<span class="id" title="var">g</span>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#MaxProps"><span class="id" title="section">MaxProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MinProps"><span class="id" title="section">MinProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="MinProps.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">G</span> <span class="id" title="var">H</span> <span class="id" title="var">M</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.solvable.gseries.html#MinProps.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="minnormal_exists"><span class="id" title="lemma">minnormal_exists</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1 <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.solvable.gseries.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.solvable.gseries.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> <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/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">M</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</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.solvable.gseries.html#MinProps.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.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.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#MinProps"><span class="id" title="section">MinProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MorphPreMax"><span class="id" title="section">MorphPreMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MorphPreMax.gT"><span class="id" title="variable">gT</span></a> <a name="MorphPreMax.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="MorphPreMax.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.solvable.gseries.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="MorphPreMax.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.solvable.gseries.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.solvable.gseries.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">Variables</span> (<a name="MorphPreMax.M"><span class="id" title="variable">M</span></a> <a name="MorphPreMax.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.solvable.gseries.html#MorphPreMax.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/>
<span class="id" title="keyword">Hypotheses</span> (<a name="MorphPreMax.dM"><span class="id" title="variable">dM</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.M"><span class="id" title="variable">M</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.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.D"><span class="id" title="variable">D</span></a>) (<a name="MorphPreMax.dG"><span class="id" title="variable">dG</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.D"><span class="id" title="variable">D</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="morphpre_maximal"><span class="id" title="lemma">morphpre_maximal</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="morphpre_maximal_eq"><span class="id" title="lemma">morphpre_maximal_eq</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#MorphPreMax.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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax.G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#MorphPreMax"><span class="id" title="section">MorphPreMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="InjmMax"><span class="id" title="section">InjmMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="InjmMax.gT"><span class="id" title="variable">gT</span></a> <a name="InjmMax.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="InjmMax.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.solvable.gseries.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="InjmMax.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.solvable.gseries.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.solvable.gseries.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">Variables</span> <a name="InjmMax.M"><span class="id" title="variable">M</span></a> <a name="InjmMax.G"><span class="id" title="variable">G</span></a> <a name="InjmMax.L"><span class="id" title="variable">L</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.solvable.gseries.html#InjmMax.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">Hypothesis</span> <a name="InjmMax.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.solvable.gseries.html#InjmMax.f"><span class="id" title="variable">f</span></a>.<br/>
<span class="id" title="keyword">Hypotheses</span> (<a name="InjmMax.dM"><span class="id" title="variable">dM</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</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.solvable.gseries.html#InjmMax.D"><span class="id" title="variable">D</span></a>) (<a name="InjmMax.dG"><span class="id" title="variable">dG</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.D"><span class="id" title="variable">D</span></a>) (<a name="InjmMax.dL"><span class="id" title="variable">dL</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.L"><span class="id" title="variable">L</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.solvable.gseries.html#InjmMax.D"><span class="id" title="variable">D</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="injm_maximal"><span class="id" title="lemma">injm_maximal</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="injm_maximal_eq"><span class="id" title="lemma">injm_maximal_eq</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.G"><span class="id" title="variable">G</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="injm_maxnormal"><span class="id" title="lemma">injm_maxnormal</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.L"><span class="id" title="variable">L</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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.L"><span class="id" title="variable">L</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="injm_minnormal"><span class="id" title="lemma">injm_minnormal</span></a> : <a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.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.solvable.gseries.html#InjmMax.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.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax.G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#InjmMax"><span class="id" title="section">InjmMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="QuoMax"><span class="id" title="section">QuoMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="QuoMax.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="QuoMax.K"><span class="id" title="variable">K</span></a> <a name="QuoMax.G"><span class="id" title="variable">G</span></a> <a name="QuoMax.H"><span class="id" title="variable">H</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.solvable.gseries.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>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="cosetpre_maximal"><span class="id" title="lemma">cosetpre_maximal</span></a> (<span class="id" title="var">Q</span> <span class="id" title="var">R</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.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Q"><span class="id" title="variable">Q</span></a>) (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#R"><span class="id" title="variable">R</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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#R"><span class="id" title="variable">R</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="cosetpre_maximal_eq"><span class="id" title="lemma">cosetpre_maximal_eq</span></a> (<span class="id" title="var">Q</span> <span class="id" title="var">R</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.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Q"><span class="id" title="variable">Q</span></a>) (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#R"><span class="id" title="variable">R</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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#R"><span class="id" title="variable">R</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="quotient_maximal"><span class="id" title="lemma">quotient_maximal</span></a> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="quotient_maximal_eq"><span class="id" title="lemma">quotient_maximal_eq</span></a> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.K"><span class="id" title="variable">K</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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="maximalJ"><span class="id" title="lemma">maximalJ</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maximal"><span class="id" title="definition">maximal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maximal_eqJ"><span class="id" title="lemma">maximal_eqJ</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.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.solvable.gseries.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maximal_eq"><span class="id" title="definition">maximal_eq</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax.H"><span class="id" title="variable">H</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#QuoMax"><span class="id" title="section">QuoMax</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="MaxNormalProps"><span class="id" title="section">MaxNormalProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="MaxNormalProps.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</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.solvable.gseries.html#MaxNormalProps.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="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">L</span> <span class="id" title="var">M</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.solvable.gseries.html#MaxNormalProps.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="maxnormal_normal"><span class="id" title="lemma">maxnormal_normal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxnormal_proper"><span class="id" title="lemma">maxnormal_proper</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#C"><span class="id" title="variable">C</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.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxnormal_sub"><span class="id" title="lemma">maxnormal_sub</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#C"><span class="id" title="variable">C</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.solvable.gseries.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.solvable.gseries.html#B"><span class="id" title="variable">B</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="ex_maxnormal_ntrivg"><span class="id" title="lemma">ex_maxnormal_ntrivg</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1<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.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</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.solvable.gseries.html#MaxNormalProps.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#N"><span class="id" title="variable">N</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxnormalM"><span class="id" title="lemma">maxnormalM</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#3ead863f625b94b2207b999cfcc823ba"><span class="id" title="notation">:&lt;&gt;:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.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.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="maxnormal_minnormal"><span class="id" title="lemma">maxnormal_minnormal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">L</span> <span class="id" title="var">M</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.html#L"><span class="id" title="variable">L</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.solvable.gseries.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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#L"><span class="id" title="variable">L</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/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="minnormal_maxnormal"><span class="id" title="lemma">minnormal_maxnormal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">L</span> <span class="id" title="var">M</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#L"><span class="id" title="variable">L</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.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#L"><span class="id" title="variable">L</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#MaxNormalProps"><span class="id" title="section">MaxNormalProps</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Simple"><span class="id" title="section">Simple</span></a>.<br/>

<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="simpleP"><span class="id" title="lemma">simpleP</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</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.solvable.gseries.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>) :<br/>
&nbsp;&nbsp;<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.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#36625695d37b6869c156bfcdf13834f7"><span class="id" title="notation">:!=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <span class="id" title="keyword">∀</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.solvable.gseries.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 class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="quotient_simple"><span class="id" title="lemma">quotient_simple</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#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.solvable.gseries.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>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="isog_simple"><span class="id" title="lemma">isog_simple</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">G</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.solvable.gseries.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>) (<span class="id" title="var">M</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.solvable.gseries.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#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.solvable.gseries.html#M"><span class="id" title="variable">M</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.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#M"><span class="id" title="variable">M</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="simple_maxnormal"><span class="id" title="lemma">simple_maxnormal</span></a> <span class="id" title="var">gT</span> (<span class="id" title="var">G</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.solvable.gseries.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 class="idref" href="mathcomp.solvable.gseries.html#simple"><span class="id" title="definition">simple</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#maxnormal"><span class="id" title="definition">maxnormal</span></a> 1 <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#Simple"><span class="id" title="section">Simple</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Chiefs"><span class="id" title="section">Chiefs</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="Chiefs.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">G</span> <span class="id" title="var">H</span> <span class="id" title="var">U</span> <span class="id" title="var">V</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.solvable.gseries.html#Chiefs.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="chief_factor_minnormal"><span class="id" title="lemma">chief_factor_minnormal</span></a> <span class="id" title="var">G</span> <span class="id" title="var">V</span> <span class="id" title="var">U</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#chief_factor"><span class="id" title="definition">chief_factor</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</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.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="acts_irrQ"><span class="id" title="lemma">acts_irrQ</span></a> <span class="id" title="var">G</span> <span class="id" title="var">U</span> <span class="id" title="var">V</span> :<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.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.solvable.gseries.html#V"><span class="id" title="variable">V</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.solvable.gseries.html#V"><span class="id" title="variable">V</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</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/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.fingroup.action.html#acts_irreducibly"><span class="id" title="definition">acts_irreducibly</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a>) <a class="idref" href="mathcomp.fingroup.action.html#a18f700a73b74a8543514202563c13c4"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#a18f700a73b74a8543514202563c13c4"><span class="id" title="notation">Q</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.solvable.gseries.html#minnormal"><span class="id" title="definition">minnormal</span></a> (<a class="idref" href="mathcomp.solvable.gseries.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a>) (<a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#V"><span class="id" title="variable">V</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="chief_series_exists"><span class="id" title="lemma">chief_series_exists</span></a> <span class="id" title="var">H</span> <span class="id" title="var">G</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.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.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">s</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.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.solvable.gseries.html#34031cd5d91f4c8b27b08f87cc5b5aaf"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#34031cd5d91f4c8b27b08f87cc5b5aaf"><span class="id" title="notation">chief</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">).-</span></a><a class="idref" href="mathcomp.solvable.gseries.html#6efcb060d1407a4d44d9b1cb19cf1e04"><span class="id" title="notation">series</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</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">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#last"><span class="id" title="definition">last</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="mathcomp.solvable.gseries.html#s"><span class="id" title="variable">s</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.solvable.gseries.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.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#Chiefs"><span class="id" title="section">Chiefs</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Central"><span class="id" title="section">Central</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="Central.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="Central.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.solvable.gseries.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>).<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">H</span> <span class="id" title="var">K</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.solvable.gseries.html#Central.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="central_factor_central"><span class="id" title="lemma">central_factor_central</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.solvable.gseries.html#central_factor"><span class="id" title="definition">central_factor</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Central.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.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">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#Central.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="central_central_factor"><span class="id" title="lemma">central_central_factor</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.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">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">Z</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.solvable.gseries.html#Central.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.solvable.center.html#07d637974acf808c1caadc3b5bdfa6d3"><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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</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.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation">&lt;|</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Central.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.solvable.gseries.html#central_factor"><span class="id" title="definition">central_factor</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#Central.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.solvable.gseries.html#K"><span class="id" title="variable">K</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.solvable.gseries.html#Central"><span class="id" title="section">Central</span></a>.<br/>
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