GAP (XMod) - Chapter 6: Actors of 2d-groups

The actor of \(\calX\) is a crossed module \(\Act(\calX) = (\Delta : \calW(\calX) \to \Aut(\calX))\) which was shown by Lue and Norrie, in [Nor87] and [Nor90] to give the automorphism object of a crossed module \(\calX\). In this implementation, the source of the actor is a permutation representation \(W\) of the Whitehead group of regular derivations, and the range of the actor is a permutation representation \(A\) of the automorphism group \(\Aut(\calX)\) of \(\calX\).

6.1-1 AutomorphismPermGroup

The automorphisms \(( \sigma, \rho )\) of \(\calX\) form a group \(\Aut(\calX)\) of crossed module isomorphisms. The function AutomorphismPermGroup finds a set of GeneratingAutomorphisms for \(\Aut(\calX)\), and then constructs a permutation representation of this group, which is used as the range of the actor crossed module of \(\calX\). The individual automorphisms can be constructed from the permutation group using the function PermAutomorphismAs2dGroupMorphism. The example below uses the crossed module X3=[c3->s3] constructed in section 5.1-1.


gap> APX3 := AutomorphismPermGroup( X3 );
Group([ (5,7,6), (1,2)(3,4)(6,7) ])
gap> Size( APX3 );
6
gap> genX3 := GeneratingAutomorphisms( X3 );    
[ [[c3->s3] => [c3->s3]], [[c3->s3] => [c3->s3]] ]
gap> e6 := Elements( APX3 )[6];
(1,2)(3,4)(5,7)
gap> m6 := PermAutomorphismAs2dGroupMorphism( X3, e6 );;
gap> Display( m6 );
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 to:
  [ (4,6,5), (2,3)(4,5) ]

The automorphisms \(( \gamma, \rho )\) of a cat\(^1\)-group \(\calC\) form a group \(\Aut(\calC)\) of cat\(^1\)-group isomorphisms. The function AutomorphismPermGroup constructs a permutation representation of this group, which is used as the range of the actor crossed module of \(\calC\). The individual automorphisms can be constructed from the permutation group using the function PermAutomorphismAs2dGroupMorphism. The example below uses the cat\(^1\)-group C3 constructed in section DerivationByImages (5.1-1).


gap> APC3 := AutomorphismPermGroup( C3 );
Group([ (1,3,2)(4,6,5)(7,9,8)(12,13,14), (2,3)(4,7)(5,9)(6,8)(10,11)(13,14) ])
gap> IdGroup( APC3 );
[ 6, 1 ]
gap> a := GeneratorsOfGroup( APC3 )[1];;
gap> m := PermAutomorphismAs2dGroupMorphism( C3, a );
[[g18 => s3] => [g18 => s3]]

6.1-2 WhiteheadXMod

An automorphism \(( \sigma, \rho )\) of \(\calX\) acts on the Whitehead monoid by \(\chi^{(\sigma,\rho)} = \sigma \circ \chi \circ \rho^{-1}\), and this determines the action for the actor. In fact the four groups \(S, W, R, A\), the homomorphisms between them, and the various actions, give five crossed modules forming a crossed square (see ActorCrossedSquare (8.2-5)).


gap> WGX3 := WhiteheadPermGroup( X3 );
Group([ (1,2,3), (1,2) ])
gap> APX3 := AutomorphismPermGroup( X3 );
Group([ (5,7,6), (1,2)(3,4)(6,7) ])
gap> WX3 := WhiteheadXMod( X3 );; 
gap> Display( WX3 );
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.
gap> LX3 := LueXMod( X3 );;
gap> Display( LX3 );
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.
gap> NX3 := NorrieXMod( X3 );; 
gap> Display( NX3 );
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.
gap> AX3 := ActorXMod( X3 );; 
gap> Display( AX3);
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.

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 XAq8 is the automorphism crossed module of the quaternion group.

 

gap> q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );;
gap> SetName( q8, "q8" );
gap> XAq8 := XModByAutomorphismGroup( q8 );; 
gap> StructureDescription( WhiteheadXMod( XAq8 ) ); 
[ "Q8", "C2 x C2 x C2" ]
gap> StructureDescription( LueXMod( XAq8 ) );      
[ "Q8", "S4" ]
gap> StructureDescription( NorrieXMod( XAq8 ) );
[ "S4", "S4" ]
gap> StructureDescription( ActorXMod( XAq8 ) ); 
[ "C2 x C2 x C2", "S4" ]

6.1-3 XModCentre

Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of \(\calW(\calX)\) and \(\calN(\calX)\) form the inner morphism of \(\calX\), mapping source elements to principal derivations and range elements to inner automorphisms. The image of \(\calX\) under this morphism is the inner actor of \(\calX\), while the kernel is the centre of \(\calX\). In the example which follows, the inner morphism of X3=(c3->s3), from Chapter 5, is an inclusion of crossed modules.

Note that we appear to have defined two sorts of centre for a crossed module: XModCentre here, and CentreXMod (4.1-7) in the chapter on isoclinism. We suspect that these two definitions give the same answer, but this remains to be resolved.


gap> IMX3 := InnerMorphism( X3 );; 
gap> Display( IMX3 );
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) ]
gap> IsInjective( IMX3 );
true
gap> ZX3 := XModCentre( X3 ); 
[Group( () )->Group( () )]
gap> IAX3 := InnerActorXMod( X3 );;  
gap> Display( IAX3 );
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.

6.2 Actor of a cat\(^1\)-group

6.2-1 ActorCat1Group

The actor of a cat\(^1\)-group \(C\) is obtained by converting \(C\) to a crossed module; forming the actor of that crossed module; and then converting that actor into a cat\(^1\)-group.


gap> C3;
[g18 => s3]
gap> AC3 := ActorCat1Group( C3 );
cat1(Actor[c3->s3])
gap> Display( AC3 );             
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]
gap> StructureDescription( AC3 );
[ "S3 x S3", "S3" ]
gap> IAC3 := InnerActorCat1Group( C3 );
cat1(InnerActor[c3->s3])
gap> StructureDescription( IAC3 );
[ "(C3 x C3) : C2", "S3" ]

6.2-2 Actor

The global functions Actor and InnerActor will call operations ActorXMod and InnerActorXMod or ActorCat1Group and InnerActorCat1Group as appropriate.