<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/pre>/div>
<h5>2.1-1 FreeGeneratorsOfGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FreeGeneratorsOfGroup</code>( <var class="Arg">u</var> )</td><tdclass="tdright">( attribute )</td></tr></table>
<p> alist of java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 117
<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</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<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> java.lang.StringIndexOutOfBoundsException: Index 255 out of bounds for length 117
<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> java.lang.StringIndexOutOfBoundsException: Index 213 out of bounds for length 74
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; IndexNC</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>return theindex of< class"rg>u2</var> <class="">u1<var, where <var =Arg>u1/> andvarclass="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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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>
<div class="func"><table class=
<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<p>Agroupofthe structure < class="">.3()31/> be obtainedasthesemidirect productspan =""G<>, ,of group <span classSimpleMath>38<span>withtheautomorphismof fieldwith< class">/ thatraises eachfield element to its fourth power. Note that the semidirect product of SU<span class="SimpleMath">(3,8)</span> with the field automorphism that squares each field element yields a group of the structure <span class="SimpleMath">3.U_3(8).6</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; \=</code>( <var class="Arg">u1</var>, <var class="Arg">u2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p> 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>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>
<anexample computes write < class="ode>f12code>in thegenerators<codeclass="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="span class="GAPprompt">gap></span> <span class="GAPinput">mats:= List( gens, m -> IdentityMat( 9, f ) );;</span>
<free group on thespan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 104
<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>
< Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 108
<span class="GAPprompt">></span> <span class="GAPinput"> GeneratorsOfGroup(g),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> GeneratorsOfGroup(u. = []</>
[ a, b ] -> [ f1^2*f2^2, f1^2*f2 ]
<span class="spanclass=""g;/><class"GAPinput"> <span
<span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( hom, f.1^2 );</span> b*a^-s ""gap>/pan> span =">:NullMat(9,9 )</span>
<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>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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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=java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<p>span class><<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 86
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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 <<p>The next step is the construction of the productproduct <span="">G/>anda , < class="SimpleMath>.3 ×()3_)/.We couldtryto the factor of span =""> 3.(831<span>modulo a diagonal subgroup of order three, by just applying the <code class="code">/</code> operation. Since <strong class="pkg">GAP</strong> would need too much time for that, and since we know better in which situation we are, we create the desired action directly on suitable sets on pairs.</p>
<p>Here is an example:</p>
<div =">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 < 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=span="APprompt"&;/span>< =>u: Image Embedding dp 1 ));<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*java.lang.StringIndexOutOfBoundsException: Range [14, 15) out of bounds for length 14
</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<spanclass=GAPprompt"gap&;/><span class"GAPinput": .3<>
<dl>
<dt><strong class="Mark">spanjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 80
<dd><p>[<span class="SimpleMath">x_1^-1, x_2, dots, x_n</span>] (<span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
</dd>
<dt><strong class="Mark">U:</strong></dt>
<dd><p>[span class=SimpleMath>_1x_2 x_2,x_3,dots x_n<span> < ""≥<)java.lang.StringIndexOutOfBoundsException: Range [110, 111) out of bounds for length 110
</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 java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 26
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