<h4>6.1 <span class="Heading">Actor of a crossed module</span></h4>
<p>The <em>actor</em> of <span class="SimpleMath">\(\calX\)</span> is a crossed module <span class="SimpleMath">\(\Act(\calX) = (\Delta : \calW(\calX) \to \Aut(\calX))\)</span> which was shown by Lue and Norrie, in <a href="chapBib_mj.html#biBN2">[Nor87]</a> and <a href="chapBib_mj.html#biBN1">[Nor90]</a> to give the automorphism object of a crossed module <span class="SimpleMath">\(\calX\)</span>. In this implementation, the source of the actor is a permutation representation <span class="SimpleMath">\(W\)</span> of the Whitehead group of regular derivations, and the range of the actor is a permutation representation <span class="SimpleMath">\(A\)</span> of the automorphism group <span class="SimpleMath">\(\Aut(\calX)\)</span> of <span class="SimpleMath">\(\calX\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismPermGroup</code>( <var class="Arg">2d-gp</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">‣ GeneratingAutomorphisms</code>( <var class="Arg">2d-gp</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">‣ PermAutomorphismAs2dGroupMorphism</code>( <var class="Arg">2d-gp</var>, <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The automorphisms <span class="SimpleMath">\(( \sigma, \rho )\)</span> of <span class="SimpleMath">\(\calX\)</span> form a group <span class="SimpleMath">\(\Aut(\calX)\)</span> of crossed module isomorphisms. The function <code class="func">AutomorphismPermGroup</code> finds a set of <code class="func">GeneratingAutomorphisms</code> for <span class="SimpleMath">\(\Aut(\calX)\)</span>, and then constructs a permutation representation of this group, which is used as the range of the actor crossed module of <span class="SimpleMath">\(\calX\)</span>. The individual automorphisms can be constructed from the permutation group using the function <code class="func">PermAutomorphismAs2dGroupMorphism</code>. The example below uses the crossed module <code class="code">X3=[c3->s3]</code> constructed in section <a href="chap5_mj.html#X83EC6F7780F5636E"><span class="RefLink">5.1-1</span></a>.</p>
<p>The automorphisms <span class="SimpleMath">\(( \gamma, \rho )\)</span> of a cat<span class="SimpleMath">\(^1\)</span>-group <span class="SimpleMath">\(\calC\)</span> form a group <span class="SimpleMath">\(\Aut(\calC)\)</span> of cat<span class="SimpleMath">\(^1\)</span>-group isomorphisms. The function <code class="func">AutomorphismPermGroup</code> constructs a permutation representation of this group, which is used as the range of the actor crossed module of <span class="SimpleMath">\(\calC\)</span>. The individual automorphisms can be constructed from the permutation group using the function <code class="func">PermAutomorphismAs2dGroupMorphism</code>. The example below uses the cat<span class="SimpleMath">\(^1\)</span>-group <code class="code">C3</code> constructed in section <code class="func">DerivationByImages</code> (<a href="chap5_mj.html#X83EC6F7780F5636E"><span class="RefLink">5.1-1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadXMod</code>( <var class="Arg">xmod</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">‣ LueXMod</code>( <var class="Arg">xmod</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">‣ NorrieXMod</code>( <var class="Arg">xmod</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">‣ ActorXMod</code>( <var class="Arg">xmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>An automorphism <span class="SimpleMath">\(( \sigma, \rho )\)</span> of <span class="SimpleMath">\(\calX\)</span> acts on the Whitehead monoid by <span class="SimpleMath">\(\chi^{(\sigma,\rho)} = \sigma \circ \chi \circ \rho^{-1}\)</span>, and this determines the action for the actor. In fact the four groups <span class="SimpleMath">\(S, W, R, A\)</span>, the homomorphisms between them, and the various actions, give five crossed modules forming a <em>crossed square</em> (see <code class="func">ActorCrossedSquare</code> (<a href="chap8_mj.html#X833362FE87ED3C48"><spanclass="RefLink">8.2-5</span></a>)).</p>
<ul>
<li><p><span class="SimpleMath">\(\calW(\calX) = (\eta : S \to W),~\)</span> the Whitehead crossed module of <span class="SimpleMath">\(\calX\)</span>, at the top,</p>
</li>
<li><p><span class="SimpleMath">\(\calX = (\partial : S \to R),~\)</span> the initial crossed module, on the left,</p>
</li>
<li><p><span class="SimpleMath">\(\Act(\calX) = ( \Delta : W \to A),~\)</span> the actor crossed module of <span class="SimpleMath">\(\calX\)</span>, on the right,</p>
</li>
<li><p><span class="SimpleMath">\(\calN(X) = (\alpha : R \to A),~\)</span> the Norrie crossed module of <span class="SimpleMath">\(\calX\)</span>, on the bottom, and</p>
</li>
<li><p><span class="SimpleMath">\(\calL(\calX) = (\Delta\circ\eta = \alpha\circ\partial : S \to A),~\)</span> the Lue crossed module of <span class="SimpleMath">\(\calX\)</span>, along the top-left to bottom-right diagonal.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WGX3 := WhiteheadPermGroup( X3 );</span>
Group([ (1,2,3), (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">APX3 := AutomorphismPermGroup( X3 );</span>
Group([ (5,7,6), (1,2)(3,4)(6,7) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">WX3 := WhiteheadXMod( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( WX3 );</span>
Crossed module Whitehead[c3->s3] :-
: Source group has generators:
[ (1,2,3)(4,6,5) ]
: Range group has generators:
[ (1,2,3), (1,2) ]
: Boundary homomorphism maps source generators to:
[ (1,2,3) ]
: Action homomorphism maps range generators to automorphisms:
(1,2,3) --> { source gens --> [ (1,2,3)(4,6,5) ] }
(1,2) --> { source gens --> [ (1,3,2)(4,5,6) ] }
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap></span> <span class="GAPinput">LX3 := LueXMod( X3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( LX3 );</span>
Crossed module Lue[c3->s3] :-
: Source group has generators:
[ (1,2,3)(4,6,5) ]
: Range group has generators:
[ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
[ (5,7,6) ]
: Action homomorphism maps range generators to automorphisms:
(5,7,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
(1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,5,6) ] }
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap></span> <span class="GAPinput">NX3 := NorrieXMod( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( NX3 );</span>
Crossed module Norrie[c3->s3] :-
: Source group has generators:
[ (4,5,6), (2,3)(5,6) ]
: Range group has generators:
[ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
[ (5,6,7), (1,2)(3,4)(6,7) ]
: Action homomorphism maps range generators to automorphisms:
(5,7,6) --> { source gens --> [ (4,5,6), (2,3)(4,5) ] }
(1,2)(3,4)(6,7) --> { source gens --> [ (4,6,5), (2,3)(5,6) ] }
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap></span> <span class="GAPinput">AX3 := ActorXMod( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( AX3);</span>
Crossed module Actor[c3->s3] :-
: Source group has generators:
[ (1,2,3), (1,2) ]
: Range group has generators:
[ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
[ (5,7,6), (1,2)(3,4)(6,7) ]
: Action homomorphism maps range generators to automorphisms:
(5,7,6) --> { source gens --> [ (1,2,3), (2,3) ] }
(1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2), (1,2) ] }
These 2 automorphisms generate the group of automorphisms.
</pre></div>
<p>The main methods for these operations are written for permutation crossed modules. For other crossed modules an isomorphism to a permutation crossed module is found first, and then the main method is applied to the image. In the example the crossed module <code class="code">XAq8</code> is the automorphism crossed module of the quaternion group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XModCentre</code>( <var class="Arg">xmod</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">‣ InnerActorXMod</code>( <var class="Arg">xmod</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">‣ InnerMorphism</code>( <var class="Arg">xmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of <span class="SimpleMath">\(\calW(\calX)\)</span> and <span class="SimpleMath">\(\calN(\calX)\)</span> form the <em>inner morphism</em> of <span class="SimpleMath">\(\calX\)</span>, mapping source elements to principal derivations and range elements to inner automorphisms. The image of <span class="SimpleMath">\(\calX\)</span> under this morphism is the <em>inner actor</em> of <span class="SimpleMath">\(\calX\)</span>, while the kernel is the <em>centre</em> of <span class="SimpleMath">\(\calX\)</span>. In the example which follows, the inner morphism of <code class="code">X3=(c3->s3)</code>, from Chapter <a href="chap5_mj.html#X85CD9A43847AE1B8"><span class="RefLink">5</span></a>, is an inclusion of crossed modules.</p>
<p>Note that we appear to have defined <em>two</em> sorts of <em>centre</em> for a crossed module: <code class="func">XModCentre</code> here, and <code class="func">CentreXMod</code> (<a href="chap4_mj.html#X7B57446086BA1BF0"><span class="RefLink">4.1-7</span></a>) in the chapter on isoclinism. We suspect that these two definitions give the same answer, but this remains to be resolved.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IMX3 := InnerMorphism( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( IMX3 );</span>
Morphism of crossed modules :-
: Source = [c3->s3] with generating sets:
[ (1,2,3)(4,6,5) ]
[ (4,5,6), (2,3)(5,6) ]
: Range = Actor[c3->s3] with generating sets:
[ (1,2,3), (1,2) ]
[ (5,7,6), (1,2)(3,4)(6,7) ]
: Source Homomorphism maps source generators to:
[ (1,2,3) ]
: Range Homomorphism maps range generators to:
[ (5,6,7), (1,2)(3,4)(6,7) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInjective( IMX3 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ZX3 := XModCentre( X3 ); </span>
[Group( () )->Group( () )]
<span class="GAPprompt">gap></span> <span class="GAPinput">IAX3 := InnerActorXMod( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( IAX3 );</span>
Crossed module InnerActor[c3->s3] :-
: Source group has generators:
[ (1,2,3) ]
: Range group has generators:
[ (5,6,7), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
[ (5,7,6) ]
: Action homomorphism maps range generators to automorphisms:
(5,6,7) --> { source gens --> [ (1,2,3) ] }
(1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2) ] }
These 2 automorphisms generate the group of automorphisms.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActorCat1Group</code>( <var class="Arg">cat1</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">‣ InnerActorCat1Group</code>( <var class="Arg">cat1</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The actor of a cat<span class="SimpleMath">\(^1\)</span>-group <span class="SimpleMath">\(C\)</span> is obtained by converting <span class="SimpleMath">\(C\)</span> to a crossed module; forming the actor of that crossed module; and then converting that actor into a cat<span class="SimpleMath">\(^1\)</span>-group.</p>
<p>A similar procedure is followed for the inner actor.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3;</span>
[g18 => s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">AC3 := ActorCat1Group( C3 );</span>
cat1(Actor[c3->s3])
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( AC3 ); </span>
Cat1-group cat1(Actor[c3->s3]) :-
: Source group has generators:
[ ( 9,10), ( 8, 9,10), ( 5, 7, 6)( 8, 9,10), (1,2)(3,4)(6,7)(8,9) ]
: Range group has generators:
[ (5,7,6), (1,2)(3,4)(6,7) ]
: tail homomorphism maps source generators to:
[ (), (), (5,7,6), (1,2)(3,4)(6,7) ]
: head homomorphism maps source generators to:
[ (1,2)(3,4)(5,6), (5,7,6), (5,7,6), (1,2)(3,4)(6,7) ]
: range embedding maps range generators to:
[ ( 5, 7, 6)( 8, 9,10), (1,2)(3,4)(6,7)(8,9) ]
: kernel has generators:
[ ( 9,10), ( 8, 9,10) ]
: boundary homomorphism maps generators of kernel to:
[ (1,2)(3,4)(5,6), (5,7,6) ]
: kernel embedding maps generators of kernel to:
[ ( 9,10), ( 8, 9,10) ]
: associated crossed module is Actor[c3->s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( AC3 );</span>
[ "S3 x S3", "S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IAC3 := InnerActorCat1Group( C3 );</span>
cat1(InnerActor[c3->s3])
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( IAC3 );</span>
[ "(C3 x C3) : C2", "S3" ]
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