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<div class="ChapSects"><a href="chap6.html#X84C872BB7F1E5F25">6 <span class="Heading">Actors of 2d-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7B853602873FC7AB">6.1 <span class="Heading">Actor of a crossed module</span></a>
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<h3>6 <span class="Heading">Actors of 2d-groups</span></h3>
<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) = (∆ : calW(calX) -> Aut(calX))</span> which was shown by Lue and Norrie, in <a href="chapBib.html#biBN2">[Nor87]</a> and <a href="chapBib.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">( σ, ρ )</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.html#X83EC6F7780F5636E"><span class="RefLink">5.1-1</span></a>.</p>
<p>The automorphisms <span class="SimpleMath">( γ, ρ )</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.html#X83EC6F7780F5636E"><span class="RefLink">5.1-1</span></a>).</p>
<=java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
<div class="func"<
<div class="func"><table class="func" [ (4,6,5, (,)(4,) ]
<divclass"func">tableclass=><td="dleft>code =func>&22 ActorXMod<code>(<var class="Arg">xmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>An automorphism <span class="SimpleMath">( ρ )</span> of <span class="SimpleMath">calX</span> acts on the Whitehead monoid by <span class="SimpleMath">χ^σ,ρ σ ^1/,andthis the for theactor In < ",,, <span the between them, andthe actions,give crossedmodules forminga<>crossed square/> see< class=func>ActorCrossedSquarecode < href".html#33362FE87ED3C48>spanclass=""8.-<></)./p>
<ul>
<li><p><span class="SimpleMath">calW(calX) = (η : S -> W),~</span> the Whitehead crossed module of <span class="SimpleMath">calX</span>, at the top,</p>
</li>
<li><p><span class="SimpleMath">calX = (
</li>
<li><p><span class="SimpleMath">Act(calX) = ( ∆ : W -> A),~</span> the actor crossed module of <span class="SimpleMath">calX</span>, on the right,</p>
/
><span class="SimpleMath">calN(X) = (α : R -> 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) = (∆∘η = α∘∂ : S -> 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<p>< id=X790EBC7C7D320C03" name"X790EBC7C7D320C03<a<p
: Range group<56.-2WhiteheadXMod</h5>
[ (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 --gt [ (13,)(45,)] java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap></span> <span class="GAPinput">LX3 := LueXMod( X3 );;</span>
< =GAPprompt>&;/ span =GAPinput"> ;span
Crossed module Lue[c3->s3] :-
: Source group has
[ (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.
< =GAPpromptgap&;/span> < class"GAPinput">X3: NorrieXMod(X3 );<span
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( NX3 );</span>
Crossedmodule Norrie[c3->3]:
: Source group has generators:
45,) 23)(56
: Range l><>< class"SimpleMath">(calX : W-gt A)~</span> the crossed moduleof spanclass=SimpleMath"calX/span> onthe right<p
(,,6) (,)(3)67) ]
: 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,,6), (23)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] :-
Sourcegroup 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) ]
: oup[ (,23,(,))
(5,7,6) --> { source gens --> [ (1,2,3), (2,3) ] }
(1,2)(3,4)(6,7) --> ><class="GAPinput">WX3 := WhiteheadXMod( X3 );; </span>
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 crossedmodule foundfirst " ;/java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
<div class="example"><pre>
<panclass=GAPprompt>&t<span spanclass="">8:=Group (12,,),,,) 1537(2,,));<spanjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
<span=GAPprompt>>/span < class="GAPinput"SetName ,"")<span>
<span class="GAPprompt">gap></span> <spanCrossed moduleLuec3-gt;s3]:
<span class="GAPprompt[(,,)465 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
[ "Q8", "C2 x C2 x C2 mapssourcegenerators :
<span class="GAPprompt">gap></span> Action maps generators automorphismsjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
[ Q8,"S4" java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<span class="GAPprompt">gap></span> <span<panclass"GAPpromptgap> );</span>
[ "S4", "S4" ]
<span class="GAPprompt">gap><java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
[ "C2:Boundary homomorphism mapssource to:
<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 (,,) --gt { -> 45) (23)(45) java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
<div class="func"><table class="func"< class"">gap;</><span =GAPinput"AX3:= (X3 ; /span
<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 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>
<p>Note that we appear to have defined <em>two</em> sorts of <em>centre</em> for a crossed module <codeclass="func"XModCentrecode here and<code/<tdclass=tdright>nbspattribute )</td></tr></table></div>
<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 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.html#X85CD9A43847AE1B8"><span class="RefLink">5</span></a>, is an inclusion of crossed modules.</p>
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
[ (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>
[( () ))-gt;/span> <span class="GAPinput">IsInjective( IMX3 );</span>
<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,java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
: 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.
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