<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
<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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">APX3 := AutomorphismPermGroup( X3 );</span>
Group([ (5,7,6), (1,2)(3,4)(6
<span class="
6 class"example"<>
<span class=
[java.lang.StringIndexOutOfBoundsException: Range [104, 2) out of bounds for length 104
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
(1,2)(3,4)(5,7)
<span class="GAPprompt">gap></span> <span class="GAPinput">m6 := PermAutomorphismAs2dGroupMorphism( X3, e6 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m6 );</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 =Source
: Source Homomorphism maps source generators to:
[ (1,3,2)(4,5,6) ]
: Range Homomorphism maps range generators"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>
4)(,45
</pre> =func< ="func" width="100%"><tr><< class""> class""&82;ActorXMod<( σ, ρ )</span> of <span class="SimpleMath">calX</span> acts on the Whitehead monoid by <span class="SimpleMath">χ^(σ,ρ) = σ ∘ χ ∘ ρ^-1</span>, and this determines the action for theσ, (ρ =σ∘ χ∘ρ^-1<span> and determinesthe action the . fact the fourgroups span class=SimpleMath">S, W R A/span>, thehomomorphisms various,givefive modules a em>crossedsquare<em( codeclass""></> (ahref=chap8#8">< classRefLink">.-<spana>)<>
<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
</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>
<: Source has:
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WGX3 := WhiteheadPermGroup( X3 );</span>
(123,12 ]java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
<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">ap><span> <span class isfound , 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>
<span class="GAPprompt">gap></span> <span lass="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 --&
These 2 automorphisms generate the group of automorphisms.
<span class="GAPprompt">gap< "">gap&;/>< class=GAPinput> := ((,,34(5,76,(,,,)2648 ;<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display<span class="gap<span> spanclass="GAPinput">( q8,q8 ;/>
Crossedmodule Lue[> -
: Source group has generators:
123(4,)]
: Range group has generators:
[ (5,7,6), (1,2)(3,4)(6,7) ]
sm maps source generatorseneratorsto
[ (5,7,6) ]
:Actionhomomorphismmaps rangegenerators to automorphisms:
(5,7,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
(1,2)(3,4)(6,7) --> { source "" S4"]
These 2 automorphisms generate the group of automorphisms.
< ="">></span> <span class="GAPinput">NX3 := NorrieXMod( X3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( NX3java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
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) ]
homomorphism generators:
[ (5,6
: Action homomorphism maps range generators to automorphisms:
(76 -gt;{sourcegens-&;[(456)(,)(,)]}
(1,2)(3,4)(6,7) --> { source gens --> [ (4,6,5), (2,3)(5,6) ] }
These 2 automorphisms generate the group of automorphisms.
span =GAPprompt>></pan span class"> :=ActorXMod );<>
<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 : class"func></> ,and <code class="func">CentreXMod</code> (<a href="chap4.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="func"><table class="func" width="100%"><tr><td )</td><td ""(&;>=GAPinput );</span>
<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 java.lang.StringIndexOutOfBoundsException: Range [399, 297) out of bounds for length 816
<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.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) ]
Group -Group( () )]
: 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><java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
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:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
: Boundary homomorphism maps source generators to:
[ (5,7,6) ]
: Action homomorphism maps range generators to automorphisms:
(5,6,7) --> { source gens --> [ (1,2,3) ] }
12(,4(7 &; { sourcegens->[1 ]
These 2 automorphisms generate the group of automorphisms.
div=>tableclassfunc"width"10%>tr>td ""< =">#27 </code(""cat1/> <td> ="">&; )<>tr>>/>/>
<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 is the inner actor.</p>
<div class="example"><pre>
<classGAPprompt>&<span> spanclass""C3;/pan
[g18 => s3]
<span class="GAPprompt">gap></span> <span class="GAPinput">AC3 := ActorCat1Group( C3 );</span>
: to
< (,7, 6)( 8, 9,10), (1,2)(3,4)(6,7)(8,9) ]
Cat1-group cat1(Actor[c3->s3]) :-
:Source group has generators
[ ( 9,10), ( 8, 9,10), ( 5, 7,
: Range group has generators:
[ (5[ "(3 xC3 C2" ""]
: tail homomorphism maps source generators to:
[ (), (), (5,7,6), (1,2)(3,4)(6,7) ]
: head homomorphism maps source generators to:
[ java.lang.StringIndexOutOfBoundsException: Index 220 out of bounds for length 220
: 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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