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<p><a id="X810FFB1C8035C8BE" name="X810FFB1C8035C8BE"></a></p>
<div class="ChapSects"><a href="chap14_mj.html#X810FFB1C8035C8BE">14 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14_mj.html#X7C9734B880042C73">14.1 <span class="Heading">Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X7F8E297F7C84DE51">14.1-1 InclusionMappingGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X81D29E737F3D4878">14.1-2 InnerAutomorphismsByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X7FC631B786C1DC8B">14.1-3 IsGroupOfAutomorphisms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14_mj.html#X852BD9CA84C2AFF0">14.2 <span class="Heading">Abelian Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X806DEFCC859BB4F1">14.2-1 AbelianModuleObject</a></span>
</div></div>
</div>

<h3>14 <span class="Heading">Utility functions</span></h3>

<p>By a utility function we mean a <strong class="pkg">GAP</strong> function which is</p>


<ul>
<li><p>needed by other functions in this package,</p>

</li>
<li><p>not (as far as we know) provided by the standard <strong class="pkg">GAP</strong> library,</p>

</li>
<li><p>more suitable for inclusion in the main library than in this package.</p>

</li>
</ul>
<p>Sections on <em>Printing Lists</em> and <em>Distinct and Common Representatives</em> were moved to the <strong class="pkg">Utils</strong> package with version 2.56.</p>

<p><a id="X7C9734B880042C73" name="X7C9734B880042C73"></a></p>

<h4>14.1 <span class="Heading">Mappings</span></h4>

<p>The following two functions have been moved to the <strong class="pkg">gpd</strong> package, but are still documented here.</p>

<p><a id="X7F8E297F7C84DE51" name="X7F8E297F7C84DE51"></a></p>

<h5>14.1-1 InclusionMappingGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InclusionMappingGroups</code>( <var class="Arg">G</var>, <var class="Arg">H</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">‣ MappingToOne</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This set of utilities concerns mappings. The map <code class="code">incd8</code> is the inclusion of <code class="code">d8</code> in <code class="code">d16</code> used in Section <a href="chap3_mj.html#X7B09A28579707CAF"><span class="RefLink">3.4</span></a>. <code class="code">MappingToOne(G,H)</code> maps the whole of <span class="SimpleMath">\(G\)</span> to the identity element in <span class="SimpleMath">\(H\)</span>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Print( incd8, "\n" );</span>
[ (11,13,15,17)(12,14,16,18), (11,18)(12,17)(13,16)(14,15) ] ->
[ (11,13,15,17)(12,14,16,18), (11,18)(12,17)(13,16)(14,15) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">imd8 := Image( incd8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MappingToOne( c4, imd8 );</span>
[ (11,13,15,17)(12,14,16,18) ] -> [ () ]

</pre></div>

<p><a id="X81D29E737F3D4878" name="X81D29E737F3D4878"></a></p>

<h5>14.1-2 InnerAutomorphismsByNormalSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InnerAutomorphismsByNormalSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Inner automorphisms of a group <code class="code">G</code> by the elements of a normal subgroup <code class="code">N</code> are calculated, often with <code class="code">G</code> = <code class="code">N</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">autd8 := AutomorphismGroup( d8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">innd8 := InnerAutomorphismsByNormalSubgroup( d8, d8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( innd8 );</span>
[ ^(1,2,3,4), ^(1,3) ]

</pre></div>

<p><a id="X7FC631B786C1DC8B" name="X7FC631B786C1DC8B"></a></p>

<h5>14.1-3 IsGroupOfAutomorphisms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupOfAutomorphisms</code>( <var class="Arg">A</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Tests whether the elements of a group are automorphisms.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupOfAutomorphisms( innd8 );</span>
true

</pre></div>

<p><a id="X852BD9CA84C2AFF0" name="X852BD9CA84C2AFF0"></a></p>

<h4>14.2 <span class="Heading">Abelian Modules</span></h4>

<p><a id="X806DEFCC859BB4F1" name="X806DEFCC859BB4F1"></a></p>

<h5>14.2-1 AbelianModuleObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianModuleObject</code>( <var class="Arg">grp</var>, <var class="Arg">act</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">‣ IsAbelianModule</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianModuleGroup</code>( <var class="Arg">obj</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">‣ AbelianModuleAction</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>An abelian module is an abelian group together with a group action. These are used by the crossed module constructor <code class="func">XModByAbelianModule</code> (<a href="chap2_mj.html#X824631577864961E"><span class="RefLink">2.1-7</span></a>).</p>

<p>The resulting <code class="code">Xabmod</code> is isomorphic to the output from <code class="code">XModByAutomorphismGroup( k4 );</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">x := (6,7)(8,9);;  y := (6,8)(7,9);;  z := (6,9)(7,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">k4a := Group( x, y );;  SetName( k4a, "k4a" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens3a := [ (1,2), (2,3) ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s3a := Group( gens3a );;  SetName( s3a, "s3a" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha := GroupHomomorphismByImages( k4a, k4a, [x,y], [y,x] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">beta := GroupHomomorphismByImages( k4a, k4a, [x,y], [x,z] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">auta := Group( alpha, beta );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">acta := GroupHomomorphismByImages( s3a, auta, gens3a, [alpha,beta] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">abmod := AbelianModuleObject( k4a, acta );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Xabmod := XModByAbelianModule( abmod );</span>
[k4a->s3a]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Xabmod );</span>

Crossed module [k4a->s3a] :- 
: Source group k4a has generators:
  [ (6,7)(8,9), (6,8)(7,9) ]
: Range group s3a has generators:
  [ (1,2), (2,3) ]
: Boundary homomorphism maps source generators to:
  [ (), () ]
: Action homomorphism maps range generators to automorphisms:
  (1,2) --> { source gens --> [ (6,8)(7,9), (6,7)(8,9) ] }
  (2,3) --> { source gens --> [ (6,7)(8,9), (6,9)(7,8) ] }
  These 2 automorphisms generate the group of automorphisms.


</pre></div>


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