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<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p>
<p><a id="X7B92AEAE7ACBE77D" name="X7B92AEAE7ACBE77D"></a></p>
<div class="ChapSects"><a href="chap2.html#X7B92AEAE7ACBE77D">2 <span class="Heading">Functionality of the FGA package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86E0C3F57990A253">2.1 <span class="Heading">New operations for free groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X829FB22C7DFBEADC">2.1-1 FreeGeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D7AE4F284F17C20">2.1-2 RankOfFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7DD1809D868165AD">2.1-3 CyclicallyReducedWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7AB44D3D80E0F85A">2.2 <span class="Heading">Method installations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87B5370C7DFD401D">2.2-1 Normalizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X857DC7B085EB0539">2.2-2 RepresentativeAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A2BF4527E08803C">2.2-3 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83A0356F839C696F">2.2-4 Index</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X851069107CACF98E">2.2-5 Intersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87BDB89B7AAFE8AD"><code>2.2-6 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7839D8927E778334">2.2-7 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X806A4814806A4814"><code>2.2-8 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81D15723804771E2">2.2-9 MinimalGeneratingSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X82B7B32E783A39B0">2.3 <span class="Heading">Constructive membership test</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78917F717E5DB86A">2.3-1 AsWordLetterRepInGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7AEFC7BA80FA4CFC">2.4 <span class="Heading">Automorphism groups of free groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87677B0787B4461A">2.4-1 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F28268F850F454E">2.4-2 IsomorphismFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X85AD957084C1E38D">2.4-3 FreeGroupEndomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8173ECA579D6172B">2.4-4 FreeGroupAutomorphismsGeneratorO</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Functionality of the FGA package</span></h3>

<p>This chapter describes methods available from the <strong class="pkg">FGA</strong> package.</p>

<p>In the following, let <var class="Arg">f</var> be a free group created by <code class="code">FreeGroup(<var class="Arg">n</var>)</code>, and let <var class="Arg">u</var>, <var class="Arg">u1</var> and <var class="Arg">u2</var> be finitely generated subgroups of <var class="Arg">f</var> created by <code class="code">Group</code> or <code class="code">Subgroup</code>, or computed from some other subgroup of <var class="Arg">f</var>. Let <var class="Arg">elm</var> be an element of <var class="Arg">f</var>.</p>

<p>For example:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( 2 );                                             </span>
<free group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">u := Group( f.1^2, f.2^2, f.1*f.2 );</span>
Group([ f1^2, f2^2, f1*f2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] );</span>
Group([ f1^2, f1^4*f2^6 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">elm := f.1;</span>
f1
<span class="GAPprompt">gap></span> <span class="GAPinput">u2 := Normalizer( u, elm );</span>
Group([ f1^2 ])
</pre></div>

<p><a id="X86E0C3F57990A253" name="X86E0C3F57990A253"></a></p>

<h4>2.1 <span class="Heading">New operations for free groups</span></h4>

<p>These new operations are available for finitely generated subgroups of free groups:</p>

<p><a id="X829FB22C7DFBEADC" name="X829FB22C7DFBEADC"></a></p>

<h5>2.1-1 FreeGeneratorsOfGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGeneratorsOfGroup</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of free generators of the finitely generated subgroup <var class="Arg">u</var> of a free group.</p>

<p>The elements in this list form an N-reduced set. In addition to being a free (and thus minimal) generating set for <var class="Arg">u</var>, this means that whenever <var class="Arg">v1</var>, <var class="Arg">v2</var> and <var class="Arg">v3</var> are elements or inverses of elements of this list, then</p>


<ul>
<li><p><span class="SimpleMath"><var class="Arg">v1</var><var class="Arg">v2</var>≠1</span> implies <span class="SimpleMath">|<var class="Arg">v1</var><var class="Arg">v2</var>|≥max(|<var class="Arg">v1</var>|, |<var class="Arg">v2</var>|)</span>, and</p>

</li>
<li><p><span class="SimpleMath"><var class="Arg">v1</var><var class="Arg">v2</var>≠1</span> and <span class="SimpleMath"><var class="Arg">v2</var><var class="Arg">v3</var>≠1</span> implies <span class="SimpleMath">|<var class="Arg">v1</var><var class="Arg">v2</var><var class="Arg">v3</var>| > |<var class="Arg">v1</var>| - |<var class="Arg">v2</var>| + |<var class="Arg">v3</var>|</span></p>

</li>
</ul>
<p>hold, where <span class="SimpleMath">|.|</span> denotes the word length.</p>

<p><a id="X7D7AE4F284F17C20" name="X7D7AE4F284F17C20"></a></p>

<h5>2.1-2 RankOfFreeGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankOfFreeGroup</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rank</code>( <var class="Arg">u</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the rank of the finitely generated subgroup <var class="Arg">u</var> of a free group.</p>

<p><a id="X7DD1809D868165AD" name="X7DD1809D868165AD"></a></p>

<h5>2.1-3 CyclicallyReducedWord</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CyclicallyReducedWord</code>( <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the cyclically reduced form of <var class="Arg">elm</var>.</p>

<p><a id="X7AB44D3D80E0F85A" name="X7AB44D3D80E0F85A"></a></p>

<h4>2.2 <span class="Heading">Method installations</span></h4>

<p>This section lists operations that are already known to <strong class="pkg">GAP</strong>. <strong class="pkg">FGA</strong> installs new methods for them so that they can also be used with free groups. In cases where <strong class="pkg">FGA</strong> installs methods that are usually only used internally, user functions are shown instead.</p>

<p><a id="X87B5370C7DFD401D" name="X87B5370C7DFD401D"></a></p>

<h5>2.2-1 Normalizer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Normalizer</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Normalizer</code>( <var class="Arg">u</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The first variant returns the normalizer of the finitely generated subgroup <var class="Arg">u2</var> in <var class="Arg">u1</var>.</p>

<p>The second variant returns the normalizer of <span class="SimpleMath">⟨ <var class="Arg">elm</var> ⟩</span> in the finitely generated subgroup <var class="Arg">u</var> (see <code class="func">Normalizer</code> (<a href="../../../doc/ref/chap39_mj.html#X87B5370C7DFD401D"><span class="RefLink">Reference: Normalizer</span></a>) in the Reference Manual).</p>

<p><a id="X857DC7B085EB0539" name="X857DC7B085EB0539"></a></p>

<h5>2.2-2 RepresentativeAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeAction</code>( <var class="Arg">u</var>, <var class="Arg">d</var>, <var class="Arg">e</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsConjugate</code>( <var class="Arg">u</var>, <var class="Arg">d</var>, <var class="Arg">e</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="code">RepresentativeAction</code> returns an element <span class="SimpleMath"><var class="Arg">r</var> ∈ <var class="Arg">u</var></span>, where <var class="Arg">u</var> is a finitely generated subgroup of a free group, such that <span class="SimpleMath"><var class="Arg">d</var>^<var class="Arg">r</var>=<var class="Arg">e</var></span>, or fail, if no such <var class="Arg">r</var> exists. <var class="Arg">d</var> and <var class="Arg">e</var> may be elements or subgroups of <var class="Arg">u</var>.</p>

<p><code class="code">IsConjugate</code> returns a boolean indicating whether such an element <var class="Arg">r</var> exists.</p>

<p><a id="X7A2BF4527E08803C" name="X7A2BF4527E08803C"></a></p>

<h5>2.2-3 Centralizer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centralizer</code>( <var class="Arg">u</var>, <var class="Arg">u2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centralizer</code>( <var class="Arg">u</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the centralizer of <var class="Arg">u2</var> or <var class="Arg">elm</var> in the finitely generated subgroup <var class="Arg">u</var> of a free group.</p>

<p><a id="X83A0356F839C696F" name="X83A0356F839C696F"></a></p>

<h5>2.2-4 Index</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Index</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndexNC</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>return the index of <var class="Arg">u2</var> in <var class="Arg">u1</var>, where <var class="Arg">u1</var> and <var class="Arg">u2</var> are finitely generated subgroups of a free group. The first variant returns fail if <var class="Arg">u2</var> is not a subgroup of <var class="Arg">u1</var>, the second may return anything in this case.</p>

<p><a id="X851069107CACF98E" name="X851069107CACF98E"></a></p>

<h5>2.2-5 Intersection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var>, <var class="Arg">\dots</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the intersection of <var class="Arg">u1</var> and <var class="Arg">u2</var>, where <var class="Arg">u1</var> and <var class="Arg">u2</var> are finitely generated subgroups of a free group.</p>

<p><a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a></p>

<h5><code>2.2-6 \in</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">elm</var>, <var class="Arg">u</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>tests whether <var class="Arg">elm</var> is contained in the finitely generated subgroup <var class="Arg">u</var> of a free group.</p>

<p><a id="X7839D8927E778334" name="X7839D8927E778334"></a></p>

<h5>2.2-7 IsSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubgroup</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether <var class="Arg">u2</var> is a subgroup of <var class="Arg">u1</var>, where <var class="Arg">u1</var> and <var class="Arg">u2</var> are finitely generated subgroups of a free group.</p>

<p><a id="X806A4814806A4814" name="X806A4814806A4814"></a></p>

<h5><code>2.2-8 \=</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>test whether the two finitely generated subgroups <var class="Arg">u1</var> and <var class="Arg">u2</var> of a free group are equal.</p>

<p><a id="X81D15723804771E2" name="X81D15723804771E2"></a></p>

<h5>2.2-9 MinimalGeneratingSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalGeneratingSet</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGeneratingSet</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfGroup</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>return generating sets for the finitely generated subgroup <var class="Arg">u</var> of a free group. <code class="code">MinimalGeneratingSet</code> and <code class="code">SmallGeneratingSet</code> return the same free generators as <code class="code">FreeGeneratorsOfGroup</code>, which are in fact a minimal generating set. <code class="code">GeneratorsOfGroup</code> also returns these generators, if no other generators were stored at creation time.</p>

<p><a id="X82B7B32E783A39B0" name="X82B7B32E783A39B0"></a></p>

<h4>2.3 <span class="Heading">Constructive membership test</span></h4>

<p>It is not only possible to test whether an element is in a finitely generated subgroup of free group, this can also be done constructively. The idiomatic way to do so is by using a homomorphism.</p>

<p>Here is an example that computes how to write <code class="code">f.1^2</code> in the generators <code class="code">a=f1^2*f2^2</code> and <code class="code">b=f.1^2*f.2</code>, checks the result, and then tries to write <code class="code">f.1</code> in the same generators:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( 2 );</span>
<free group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">ua := f.1^2*f.2^2;; ub := f.1^2*f.2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u := Group( ua, ub );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := FreeGroup( "a", "b" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">hom := GroupHomomorphismByImages( g, u,</span>
<span class="GAPprompt">></span> <span class="GAPinput">            GeneratorsOfGroup(g),</span>
<span class="GAPprompt">></span> <span class="GAPinput">            GeneratorsOfGroup(u) );</span>
[ a, b ] -> [ f1^2*f2^2, f1^2*f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># how can f.1^2 be expressed?</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( hom, f.1^2 );</span>
b*a^-1*b
<span class="GAPprompt">gap></span> <span class="GAPinput">last ^ hom; # check this</span>
f1^2
<span class="GAPprompt">gap></span> <span class="GAPinput">ub * ua^-1 * ub; # another check</span>
f1^2
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( hom, f.1 ); # try f.1</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">f.1 in u;</span>
false
</pre></div>

<p>There are also lower level operations to get the same results.</p>

<p><a id="X78917F717E5DB86A" name="X78917F717E5DB86A"></a></p>

<h5>2.3-1 AsWordLetterRepInGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsWordLetterRepInGenerators</code>( <var class="Arg">elm</var>, <var class="Arg">u</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsWordLetterRepInFreeGenerators</code>( <var class="Arg">elm</var>, <var class="Arg">u</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>return a letter representation (see Section <a href="../../../doc/ref/chap37_mj.html#X80A9F39582ED296E"><span class="RefLink">Reference: Representations for Associative Words</span></a> in the <strong class="pkg">GAP</strong> Reference Manual) of the given <var class="Arg">elm</var> relative to the generators the group was created with or the free generators as returned by <code class="code">FreeGeneratorsOfGroup</code>.</p>

<p>Continuing the above example:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AsWordLetterRepInGenerators( f.1^2, u );    </span>
[ 2, -1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AsWordLetterRepInFreeGenerators( f.1^2, u );</span>
[ 2 ]
</pre></div>

<p>This means: to get <code class="code">f.1^2</code>, multiply the second of the given generators with the inverse of the first and again with the second; or just take the second free generator.</p>

<p><a id="X7AEFC7BA80FA4CFC" name="X7AEFC7BA80FA4CFC"></a></p>

<h4>2.4 <span class="Heading">Automorphism groups of free groups</span></h4>

<p>The <strong class="pkg">FGA</strong> package knows presentations of the automorphism groups of free groups. It also allows to express an automorphism as word in the generators of these presentations. This sections repeats the <strong class="pkg">GAP</strong> standard methods to do so and shows functions to obtain the generating automorphisms.</p>

<p><a id="X87677B0787B4461A" name="X87677B0787B4461A"></a></p>

<h5>2.4-1 AutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">u</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the automorphism group of the finitely generated subgroup <var class="Arg">u</var> of a free group.</p>

<p>Only a few methods will work with this group. But there is a way to obtain an isomorphic finitely presented group:</p>

<p><a id="X7F28268F850F454E" name="X7F28268F850F454E"></a></p>

<h5>2.4-2 IsomorphismFpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismFpGroup</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an isomorphism of <var class="Arg">group</var> to a finitely presented group. For automorphism groups of free groups, the <strong class="pkg">FGA</strong> package implements the presentations of <a href="chapBib.html#biBNeumann33">[Neu33]</a>. The finitely presented group itself can then be obtained with the command <code class="code">Range</code>.</p>

<p>Here is an example:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := AutomorphismGroup( f );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismFpGroup( a );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Range( iso );</span>
<fp group on the generators [ O, P, U ]>
</pre></div>

<p>To express an automorphism as word in the generators of the presentation, just apply the isomorphism obtained from <code class="code">IsomorphismFpGroup</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut := GroupHomomorphismByImages( f, f,</span>
<span class="GAPprompt">></span> <span class="GAPinput">             GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] );</span>
[ f1, f2 ] -> [ f2^-1*f1*f2, f1*f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( iso, aut );</span>
O^2*U*O*P^-1*U
</pre></div>

<p>It is also possible to use <code class="code">aut^iso</code> or <code class="code">Image( iso, aut )</code>. Using <code class="code">Image</code> will perform additional checks on the arguments.</p>

<p>The <strong class="pkg">FGA</strong> package provides a simpler way to create endomorphisms:</p>

<p><a id="X85AD957084C1E38D" name="X85AD957084C1E38D"></a></p>

<h5>2.4-3 FreeGroupEndomorphismByImages</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupEndomorphismByImages</code>( <var class="Arg">g</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the endomorphism that maps the free generators of the finitely generated subgroup <var class="Arg">g</var> of a free group to the elements listed in <var class="Arg">imgs</var>. You may then apply <code class="code">IsBijective</code> to check whether it is an automorphism.</p>

<p>The following functions return automorphisms that correspond to the generators in the presentation:</p>

<p><a id="X8173ECA579D6172B" name="X8173ECA579D6172B"></a></p>

<h5>2.4-4 FreeGroupAutomorphismsGeneratorO</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorO</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorP</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorU</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorS</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorT</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorQ</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupAutomorphismsGeneratorR</code>( <var class="Arg">group</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>return the automorphism which maps the free generators [<span class="SimpleMath">x_1, x_2, dots, x_n</span>] of <var class="Arg">group</var> to</p>


<dl>
<dt><strong class="Mark">O:</strong></dt>
<dd><p>[<span class="SimpleMath">x_1^-1, x_2, dots, x_n</span>] (<span class="SimpleMath">n≥1</span>)</p>

</dd>
<dt><strong class="Mark">P:</strong></dt>
<dd><p>[<span class="SimpleMath">x_2, x_1, x_3, dots, x_n</span>] (<span class="SimpleMath">n≥2</span>)</p>

</dd>
<dt><strong class="Mark">U:</strong></dt>
<dd><p>[<span class="SimpleMath">x_1x_2, x_2, x_3, dots, x_n</span>] (<span class="SimpleMath">n≥2</span>)</p>

</dd>
<dt><strong class="Mark">S:</strong></dt>
<dd><p>[<span class="SimpleMath">x_2^-1, x_3^-1, dots, x_n^-1, x_1^-1</span>] (<span class="SimpleMath">n≥1</span>)</p>

</dd>
<dt><strong class="Mark">T:</strong></dt>
<dd><p>[<span class="SimpleMath">x_2, x_1^-1, x_3, dots, x_n</span>] (<span class="SimpleMath">n≥2</span>)</p>

</dd>
<dt><strong class="Mark">Q:</strong></dt>
<dd><p>[<span class="SimpleMath">x_2, x_3, dots, x_n, x_1</span>] (<span class="SimpleMath">n≥2</span>)</p>

</dd>
<dt><strong class="Mark">R:</strong></dt>
<dd><p>[<span class="SimpleMath">x_2^-1, x_1, x_3, x_4, dots, x_n-2, x_nx_n-1^-1, x_n-1^-1</span>] (<span class="SimpleMath">n≥4</span>)</p>

</dd>
</dl>

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