chap6.txt

  
  6 Actors of 2d-groups
  
  
  6.1 Actor of a crossed module
  
  The  actor  of  mathcalX  is  a  crossed  module  (∆ : mathcalW(mathcalX) ->
  Aut(mathcalX))  which was shown by Lue and Norrie, in [Nor87] and [Nor90] to
  give  the  automorphism  object  of  a  crossed  module  mathcalX.  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(mathcalX) of
  mathcalX.
  
  6.1-1 AutomorphismPermGroup
  
  AutomorphismPermGroup( xmod )  attribute
  GeneratingAutomorphisms( xmod )  attribute
  PermAutomorphismAsXModMorphism( xmod, perm )  attribute
  
  The automorphisms ( σ, ρ ) of mathcalX form a group Aut(mathcalX) of crossed
  module  isomorphisms.  The  function  AutomorphismPermGroup  finds  a set of
  GeneratingAutomorphisms for Aut(mathcalX), and then constructs a permutation
  representation  of  this  group,  which  is  used  as the range of the actor
  crossed  module of mathcalX. 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.
  
    Example  
    
    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
  AutomorphismPermGroup( xmod )  attribute
  
  An  automorphism ( σ, ρ ) of X acts on the Whitehead monoid by χ^(σ,ρ) = σ ∘
  χ  ∘  ρ^-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:
  
      mathcalX = (∂ : S -> R),~ the initial crossed module, on the left,
  
      mathcalW(mathcalX)  =  (η  : S -> W),~ the Whitehead crossed module of
        mathcalX, at the top,
  
      mathcalN(X) = (α : R -> A),~ the Norrie crossed module of mathcalX, at
        the bottom,
  
      Act(mathcalX)  = ( ∆ : W -> A),~ the actor crossed module of mathcalX,
        on the right, and
  
      mathcalL(mathcalX)  = (∆∘η = α∘∂ : S -> A),~ the Lue crossed module of
        mathcalX, along the top-left to bottom-right diagonal.
  
    Example  
    
    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  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,
  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.
  
    Example  
    
    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( () )]