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<p id="mathjaxlink" class="pcenter"><a href="chap6_mj.html">[MathJax on]</a></p>
<p><a id="X82E82E45786E1B22" name="X82E82E45786E1B22"></a></p>
<div class="ChapSects"><a href="chap6.html#X82E82E45786E1B22">6 <span class="Heading">Totally and Mutually Permutable Products</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X85291E9D86736F21">6.1 <span class="Heading">Functions for Mutually and Totally Permutable Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X876F4E0D7C796134">6.1-1 AreMutuallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X80225ABC79C7875F">6.1-2 OnePairShowingNotMutuallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7D58DFE57AA4573B">6.1-3 AreTotallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7B51474D878ACD11">6.1-4 OnePairShowingNotTotallyPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X823C6D68813B3AC8">6.1-5 AreMutuallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83E0517A800AB839">6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7B9DA1C2834CA53D">6.1-7 AreTotallyFPermutableSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CD0B5737A25C1AD">6.1-8 OnePairShowingNotTotallyFPermutableSubgroups</a></span>
</div></div>
</div>

<h3>6 <span class="Heading">Totally and Mutually Permutable Products</span></h3>

<p>In recent years, many authors have considered totally and mutually permutable subgroups. Recall that two subgroups <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> of a group <span class="SimpleMath">G</span> are <em>totally permutable</em> if every subgroup of <span class="SimpleMath">A</span> permutes with every subgroup of <span class="SimpleMath">B</span>, and they are <em>mutually permutable</em> if every subgroup of <span class="SimpleMath">A</span> permutes with <span class="SimpleMath">B</span> and every subgroup of <span class="SimpleMath">B</span> permutes with <span class="SimpleMath">A</span>.</p>

<p>We have defined some "One" functions which give a pair of subgroups which do not permute and prove that two subgroups fail to have a certain property.</p>

<p>We have also defined some functions to work with totally and mutually <span class="SimpleMath">f</span>-permutable subgroups, where <span class="SimpleMath">f</span> is a subgroup embedding functor.</p>

<p>The functions of this chapter are defined in a preliminary state.</p>

<p><a id="X85291E9D86736F21" name="X85291E9D86736F21"></a></p>

<h4>6.1 <span class="Heading">Functions for Mutually and Totally Permutable Products</span></h4>

<p><a id="X876F4E0D7C796134" name="X876F4E0D7C796134"></a></p>

<h5>6.1-1 AreMutuallyPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreMutuallyPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns <code class="keyw">true</code> if the subgroups <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> of <span class="SimpleMath">G</span> are mutually permutable subgroups, that is, every subgroup of <span class="SimpleMath">A</span> permutes with <span class="SimpleMath">B</span> and every subgroup of <span class="SimpleMath">B</span> permutes with <span class="SimpleMath">A</span>, and <code class="keyw">false</code> otherwise. The method used here checks only that <span class="SimpleMath">A</span> permutes with all cyclic subgroups of <span class="SimpleMath">B</span> and that <span class="SimpleMath">B</span> permutes with all cyclic subgroups of <span class="SimpleMath">A</span>.</p>

<p>The method with two arguments assume that <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> have a common supergroup.</p>

<p><a id="X80225ABC79C7875F" name="X80225ABC79C7875F"></a></p>

<h5>6.1-2 OnePairShowingNotMutuallyPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnePairShowingNotMutuallyPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a pair of the form [ <var class="Arg">A</var>, <var class="Arg">V</var> ] with <var class="Arg">V</var> a subgroup of <var class="Arg">B</var> or of the form [ <var class="Arg">W</var>, <var class="Arg">B</var> ] with <var class="Arg">W</var> a subgroup of <var class="Arg">A</var> in which both subgroups do not permute, or <code class="keyw">fail</code> if this pair does not exist because the product is mutually permutable.</p>

<p><a id="X7D58DFE57AA4573B" name="X7D58DFE57AA4573B"></a></p>

<h5>6.1-3 AreTotallyPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreTotallyPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns <code class="keyw">true</code> if the subgroups <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> of <span class="SimpleMath">G</span> are totally permutable, that is, every subgroup of <span class="SimpleMath">A</span> permutes with every subgroup of <span class="SimpleMath">B</span>, and <code class="keyw">false</code> otherwise. The method used here checks only that every cyclic subgroup of <span class="SimpleMath">A</span> permutes with every cyclic subgroup of <span class="SimpleMath">B</span>.</p>

<p>The method with two arguments assume that <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> have a common supergroup.</p>

<p><a id="X7B51474D878ACD11" name="X7B51474D878ACD11"></a></p>

<h5>6.1-4 OnePairShowingNotTotallyPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnePairShowingNotTotallyPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a pair of the form [ <var class="Arg">V</var>, <var class="Arg">W</var> ], with <var class="Arg">V</var> a subgroup of <var class="Arg">A</var> and <var class="Arg">W</var> a subgroup of <var class="Arg">B</var>, such that both subgroups do not permute, or <code class="keyw">fail</code> if this pair does not exist because the product is totally permutable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AlternatingGroup(4);</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Subgroup(g,[(1,2,3,4),(1,3)]);</span>
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AreMutuallyPermutableSubgroups(g,a,b);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AreTotallyPermutableSubgroups(g,a,b);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OnePairShowingNotTotallyPermutableSubgroups(g,a,b);</span>
[ Group([ (2,3,4) ]), Group([ (1,2)(3,4) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=Subgroup(g,[(1,2,3)]);</span>
Group([ (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AreMutuallyPermutableSubgroups(g,a,c);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OnePairShowingNotMutuallyPermutableSubgroups(g,a,c);</span>
[ Group([ (2,3,4) ]), Group([ (1,2,3) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AreMutuallyPermutableSubgroups(a,c);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(3);</span>
Sym( [ 1 .. 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AlternatingGroup(3);</span>
Alt( [ 1 .. 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Subgroup(g,[(1,2)]);</span>
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AreTotallyPermutableSubgroups(g,a,b);</span>
true
</pre></div>

<p><a id="X823C6D68813B3AC8" name="X823C6D68813B3AC8"></a></p>

<h5>6.1-5 AreMutuallyFPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreMutuallyFPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">fA</var>, <var class="Arg">fB</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns <code class="keyw">true</code> if the subgroups <var class="Arg">A</var> and <var class="Arg">B</var> are mutually <var class="Arg">f</var>-permutable, and <code class="keyw">false</code> otherwise. Here <var class="Arg">A</var> and <var class="Arg">B</var> are subgroups of <var class="Arg">G</var> and <var class="Arg">fA</var> and <var class="Arg">fB</var> are, respectively, lists of subgroups of <var class="Arg">A</var> and <var class="Arg">B</var>, respectively.</p>

<p>In the version with four arguments, <span class="SimpleMath">A</span> and <span class="SimpleMath">B</span> are assumed to be subgroups of a common supergroup.</p>

<p><a id="X83E0517A800AB839" name="X83E0517A800AB839"></a></p>

<h5>6.1-6 OnePairShowingNotMutuallyFPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnePairShowingNotMutuallyFPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">fA</var>, <var class="Arg">fB</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a pair of the form [ <var class="Arg">A</var>, <var class="Arg">V</var> ] with <var class="Arg">V</var> a subgroup in <var class="Arg">fB</var> or <var class="Arg">B</var> or of the form [ <var class="Arg">W</var>, <var class="Arg">B</var> ] with <var class="Arg">W</var> a subgroup in <var class="Arg">fA</var> or <var class="Arg">A</var> in which both subgroups do not permute, or <code class="keyw">fail</code> if this pair does not exist. Here <var class="Arg">A</var> and <var class="Arg">B</var> are subgroups of <var class="Arg">G</var> and <var class="Arg">fA</var> and <var class="Arg">fB</var> are lists of subgroups of <var class="Arg">A</var> and <var class="Arg">B</var>, respectively.</p>

<p>In the version with four arguments, <var class="Arg">A</var> and <var class="Arg">B</var> are assumed to be subgroups of a common supergroup.</p>

<p><a id="X7B9DA1C2834CA53D" name="X7B9DA1C2834CA53D"></a></p>

<h5>6.1-7 AreTotallyFPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreTotallyFPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">fA</var>, <var class="Arg">fB</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns <code class="keyw">true</code> if the subgroup <var class="Arg">A</var> permutes with all subgroups in the list <var class="Arg">fB</var> and <var class="Arg">B</var> permutes with all subgroups in the list <var class="Arg">fA</var>, and <code class="keyw">false</code> otherwise. Here <var class="Arg">A</var> and <var class="Arg">B</var> are subgroups of <var class="Arg">G</var>, <var class="Arg">fA</var> is a list of subgroups of <var class="Arg">A</var> and <var class="Arg">fB</var> is a list of subgroups of <var class="Arg">B</var>.</p>

<p>In the version with four arguments, <var class="Arg">A</var> and <var class="Arg">B</var> are assumed to be subgroups of a common supergroup.</p>

<p><a id="X7CD0B5737A25C1AD" name="X7CD0B5737A25C1AD"></a></p>

<h5>6.1-8 OnePairShowingNotTotallyFPermutableSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnePairShowingNotTotallyFPermutableSubgroups</code>( [<var class="Arg">G</var>, ]<var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">fA</var>, <var class="Arg">fB</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a pair of the form [ <var class="Arg">U</var>, <var class="Arg">V</var> ] with <var class="Arg">U</var> a subgroup in <var class="Arg">fA</var> or <var class="Arg">A</var> and <var class="Arg">V</var> a subgroup in <var class="Arg">fB</var> or <var class="Arg">B</var> in which both subgroups do not permute, or <code class="keyw">fail</code> if this pair does not exist. Here <var class="Arg">A</var> and <var class="Arg">B</var> are subgroups of <var class="Arg">G</var>, <var class="Arg">fA</var> is a list of subgroups of <var class="Arg">A</var> and <var class="Arg">fB</var> is a list of subgroups of <var class="Arg">B</var>.</p>

<p>In the version with two arguments, <var class="Arg">A</var> and <var class="Arg">B</var> are assumed to be subgroups of a common supergroup.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=SymmetricGroup(4);</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AlternatingGroup(4);</span>
Alt( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Subgroup(g,[(1,2,3,4),(1,3)]);</span>
Group([ (1,2,3,4), (1,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AreTotallyFPermutableSubgroups(g,a,b,</span>
<span class="GAPprompt">></span> <span class="GAPinput">     MaximalSubgroups(a),MaximalSubgroups(b));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">OnePairShowingNotTotallyFPermutableSubgroups(g,a,b,</span>
<span class="GAPprompt">></span> <span class="GAPinput">     MaximalSubgroups(a),MaximalSubgroups(b));</span>
[ Group([ (1,2,3) ]), Group([ (2,4), (1,3)(2,4) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AreTotallyFPermutableSubgroups(g,a,b,DerivedSeries(a),DerivedSeries(b));</span>
true
</pre></div>


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