GAP (XMod) - Chapter 5: Actors of 2d-groups

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5 Actors of 2d-groups
5.1 Actor of a crossed module
5.1-1 WhiteheadXMod
5.1-2 XModCentre

### 5 Actors of 2d-groups

#### 5.1 Actor of a crossed module

The actor of cal X is a crossed module (∆ : cal W(cal X) -> Aut(cal X)) which was shown by Lue and Norrie, in [Nor87] and [Nor90] to give the automorphism object of a crossed module cal X. In this implementation, the source of the actor is a permutation representation W of the Whitehead group of regular derivations, and the range is a permutation representation A of the automorphism group Aut(cal X) of cal X.

##### 5.1-1 WhiteheadXMod

`‣ WhiteheadXMod`( xmod ) | ( attribute )

`‣ LueXMod`( xmod ) | ( attribute )

`‣ NorrieXMod`( xmod ) | ( attribute )

`‣ ActorXMod`( xmod ) | ( attribute )

`‣ AutomorphismPermGroup`( xmod ) | ( attribute )

An automorphism ( σ, ρ ) of `X` acts on the Whitehead monoid by χ^(σ,ρ) = σ ∘ χ ∘ ρ^-1, and this action 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:

- cal X = (∂ : S -> R),~ the initial crossed module, on the left,

- cal W(X) = (η : S -> W),~ the Whitehead crossed module of cal X, at the top,

- cal L(X) = (∆∘η = α∘∂ : S -> A),~ the Lue crossed module of cal X, along the top-left to bottom-right diagonal,

- cal N(X) = (α : R -> A),~ the Norrie crossed module of cal X, at the bottom, and

- Act(cal X) = ( ∆ : W -> A),~ the actor crossed module of cal X, on the right.

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

##### 5.1-2 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 mathcalW(mathcalX) and mathcalN(mathcalX) form the inner morphism of mathcalX, mapping source elements to principal derivations and range elements to inner automorphisms. The image of mathcalX under this morphism is the inner actor of mathcalX, while the kernel is the centre of mathcalX. In the example which follows, using the crossed module `(X3 : c3 -> s3)` from Chapter 4, the inner morphism is an inclusion of crossed modules.

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

    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:
      [ (1,3,2)(4,6,5) ]
    : 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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