<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
