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

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6 Actors of 2d-groups
6.1 Actor of a crossed module
6.1-1 AutomorphismPermGroup
6.1-2 WhiteheadXMod
6.1-3 XModCentre

### 6 Actors of 2d-groups

#### 6.1 Actor of a crossed module

The actor of \(\mathcal{X}\) is a crossed module \(\Act(\mathcal{X}) = (\Delta : \mathcal{W}(\mathcal{X}) \to \Aut(\mathcal{X}))\) which was shown by Lue and Norrie, in [Nor87] and [Nor90] to give the automorphism object of a crossed module \(\mathcal{X}\). 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 \(\Aut(\mathcal{X})\) of \(\mathcal{X}\).

##### 6.1-1 AutomorphismPermGroup

`‣ AutomorphismPermGroup`( xmod ) | ( attribute )

`‣ GeneratingAutomorphisms`( xmod ) | ( attribute )

`‣ PermAutomorphismAsXModMorphism`( xmod, perm ) | ( operation )

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

    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 := PermAutomorphismAsXModMorphism( 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) ]

##### 6.1-2 WhiteheadXMod

`‣ WhiteheadXMod`( xmod ) | ( attribute )

`‣ LueXMod`( xmod ) | ( attribute )

`‣ NorrieXMod`( xmod ) | ( attribute )

`‣ ActorXMod`( xmod ) | ( attribute )

An automorphism \(( \sigma, \rho )\) of `X` 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 \(R, S, W, A\), the homomorphisms between them, and the various actions, give five crossed modules forming a crossed square:

- \(\mathcal{X} = (\partial : S \to R),~\) the initial crossed module, on the left,

- \(\mathcal{W}(\mathcal{X}) = (\eta : S \to W),~\) the Whitehead crossed module of \(\mathcal{X}\), at the top,

- \(\mathcal{N}(X) = (\alpha : R \to A),~\) the Norrie crossed module of \(\mathcal{X}\), at the bottom,

- \(\Act(\mathcal{X}) = ( \Delta : W \to A),~\) the actor crossed module of \(\mathcal{X}\), on the right, and

- \(\mathcal{L}(\mathcal{X}) = (\Delta\circ\eta = \alpha\circ\partial : S \to A),~\) the Lue crossed module of \(\mathcal{X}\), along the top-left to bottom-right diagonal.

    gap> WGX3 := WhiteheadPermGroup( X3 );
    Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
    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)(4,5,6), (1,4)(2,6)(3,5) ]
    : Boundary homomorphism maps source generators to:
      [ (1,2,3)(4,5,6) ]
    : Action homomorphism maps range generators to automorphisms:
      (1,2,3)(4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
      (1,4)(2,6)(3,5) --> { 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)(4,5,6), (1,4)(2,6)(3,5) ]
    : 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)(4,5,6), (1,6)(2,5)(3,4) ] }
      (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,6,5), (1,4)(2,6)(3,5) ] }
      These 2 automorphisms generate the group of automorphisms.

    gap> IAX3 := InnerActorXMod( X3 );;
    gap> Display( IAX3 );
    Crossed module InnerActor[c3->s3] :-
    : Source group has generators:
      [ (1,2,3)(4,5,6) ]
    : 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)(4,5,6) ] }
      (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,6,5) ] }
      These 2 automorphisms generate the group of automorphisms.

##### 6.1-3 XModCentre

`‣ XModCentre`( xmod ) | ( attribute )

`‣ InnerActorXMod`( xmod ) | ( attribute )

`‣ InnerMorphism`( xmod ) | ( attribute )

Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of \(\mathcal{W}(\mathcal{X})\) and \(\mathcal{N}(\mathcal{X})\) form the inner morphism of \(\mathcal{X}\), mapping source elements to principal derivations and range elements to inner automorphisms. The image of \(\mathcal{X}\) under this morphism is the inner actor of \(\mathcal{X}\), while the kernel is the centre of \(\mathcal{X}\). 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)(4,5,6), (1,4)(2,6)(3,5) ]
      [ (5,7,6), (1,2)(3,4)(6,7) ]
    : Source Homomorphism maps source generators to:
      [ (1,2,3)(4,5,6) ]
    : 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( () )]

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