  
  [1X5 [33X[0;0YActors of 2d-groups[133X[101X
  
  
  [1X5.1 [33X[0;0YActor of a crossed module[133X[101X
  
  [33X[0;0YThe  [13Xactor[113X  of  [22Xcal  X[122X  is a crossed module [22X(∆ : cal W(cal X) -> Aut(cal X))[122X
  which  was  shown  by  Lue  and  Norrie,  in [Nor87] and [Nor90] to give the
  automorphism  object  of a crossed module [22Xcal X[122X. In this implementation, the
  source of the actor is a permutation representation [22XW[122X of the Whitehead group
  of  regular  derivations, and the range is a permutation representation [22XA[122X of
  the automorphism group [22XAut(cal X)[122X of [22Xcal X[122X.[133X
  
  [1X5.1-1 WhiteheadXMod[101X
  
  [29X[2XWhiteheadXMod[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XLueXMod[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XNorrieXMod[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XActorXMod[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XAutomorphismPermGroup[102X( [3Xxmod[103X ) [32X attribute
  
  [33X[0;0YAn  automorphism [22X( σ, ρ )[122X of [10XX[110X acts on the Whitehead monoid by [22Xχ^(σ,ρ) = σ ∘
  χ  ∘  ρ^-1[122X, and this action determines the action for the actor. In fact the
  four  groups  [22XR,  S,  W,  A[122X, the homomorphisms between them, and the various
  actions, give five crossed modules forming a [13Xcrossed square[113X:[133X
  
  [30X    [33X[0;6Y[22Xcal X = (∂ : S -> R)[122X,~ the initial crossed module, on the left,[133X
  
  [30X    [33X[0;6Y[22Xcal  W(X)  =  (η : S -> W)[122X,~ the Whitehead crossed module of [22Xcal X[122X, at
        the top,[133X
  
  [30X    [33X[0;6Y[22Xcal  L(X)  =  (∆∘η  = α∘∂ : S -> A)[122X,~ the Lue crossed module of [22Xcal X[122X,
        along the top-left to bottom-right diagonal,[133X
  
  [30X    [33X[0;6Y[22Xcal  N(X)  = (α : R -> A)[122X,~ the Norrie crossed module of [22Xcal X[122X, at the
        bottom, and[133X
  
  [30X    [33X[0;6Y[22XAct(cal X) = ( ∆ : W -> A)[122X,~ the actor crossed module of [22Xcal X[122X, on the
        right.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XWGX3 := WhiteheadPermGroup( X3 );[127X[104X
    [4X[28XGroup([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XAPX3 := AutomorphismPermGroup( X3 );[127X[104X
    [4X[28XGroup([ (5,7,6), (1,2)(3,4)(6,7) ])[128X[104X
    [4X[25Xgap>[125X [27XWX3 := WhiteheadXMod( X3 );; [127X[104X
    [4X[25Xgap>[125X [27XDisplay( WX3 );[127X[104X
    [4X[28XCrossed module Whitehead[c3->s3] :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ (1,2,3)(4,6,5) ][128X[104X
    [4X[28X: Range group has generators:[128X[104X
    [4X[28X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (1,3,2)(4,6,5) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (1,2,3)(4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }[128X[104X
    [4X[28X  (1,4)(2,6)(3,5) --> { source gens --> [ (1,3,2)(4,5,6) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[25Xgap>[125X [27XLX3 := LueXMod( X3 );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( LX3 );[127X[104X
    [4X[28XCrossed module Lue[c3->s3] :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ (1,2,3)(4,6,5) ][128X[104X
    [4X[28X: Range group has generators:[128X[104X
    [4X[28X  [ (5,7,6), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,7,6) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (5,7,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }[128X[104X
    [4X[28X  (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,5,6) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[25Xgap>[125X [27XNX3 := NorrieXMod( X3 );; [127X[104X
    [4X[25Xgap>[125X [27XDisplay( NX3 );[127X[104X
    [4X[28XCrossed module Norrie[c3->s3] :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X: Range group has generators:[128X[104X
    [4X[28X  [ (5,7,6), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,6,7), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (5,7,6) --> { source gens --> [ (4,5,6), (2,3)(4,5) ] }[128X[104X
    [4X[28X  (1,2)(3,4)(6,7) --> { source gens --> [ (4,6,5), (2,3)(5,6) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[25Xgap>[125X [27XAX3 := ActorXMod( X3 );;  Display( AX3);[127X[104X
    [4X[28XCrossed module Actor[c3->s3] :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][128X[104X
    [4X[28X: Range group has generators:[128X[104X
    [4X[28X  [ (5,7,6), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,6,7), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (5,7,6) --> { source gens --> [ (1,2,3)(4,5,6), (1,5)(2,4)(3,6) ] }[128X[104X
    [4X[28X  (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,6,5), (1,4)(2,6)(3,5) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.1-2 XModCentre[101X
  
  [29X[2XXModCentre[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XInnerActorXMod[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XInnerMorphism[102X( [3Xxmod[103X ) [32X attribute
  
  [33X[0;0YPairs  of  boundaries  or identity mappings provide six morphisms of crossed
  modules.   In   particular,   the   boundaries   of  [22XmathcalW(mathcalX)[122X  and
  [22XmathcalN(mathcalX)[122X  form  the  [13Xinner  morphism[113X  of  [22XmathcalX[122X, mapping source
  elements to principal derivations and range elements to inner automorphisms.
  The  image  of  [22XmathcalX[122X under this morphism is the [13Xinner actor[113X of [22XmathcalX[122X,
  while  the  kernel  is the [13Xcentre[113X of [22XmathcalX[122X. In the example which follows,
  using  the crossed module [10X(X3 : c3 -> s3)[110X from Chapter [14X4[114X, the inner morphism
  is an inclusion of crossed modules.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIAX3 := InnerActorXMod( X3 );;  [127X[104X
    [4X[25Xgap>[125X [27XDisplay( IAX3 );[127X[104X
    [4X[28XCrossed module InnerActor[c3->s3] :- [128X[104X
    [4X[28X: Source group has generators:[128X[104X
    [4X[28X  [ (1,3,2)(4,6,5) ][128X[104X
    [4X[28X: Range group has generators:[128X[104X
    [4X[28X  [ (5,6,7), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Boundary homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (5,7,6) ][128X[104X
    [4X[28X: Action homomorphism maps range generators to automorphisms:[128X[104X
    [4X[28X  (5,6,7) --> { source gens --> [ (1,3,2)(4,6,5) ] }[128X[104X
    [4X[28X  (1,2)(3,4)(6,7) --> { source gens --> [ (1,2,3)(4,5,6) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIMX3 := InnerMorphism( X3 );; [127X[104X
    [4X[25Xgap>[125X [27XDisplay( IMX3 );[127X[104X
    [4X[28XMorphism of crossed modules :- [128X[104X
    [4X[28X: Source = [c3->s3] with generating sets:[128X[104X
    [4X[28X  [ (1,2,3)(4,6,5) ][128X[104X
    [4X[28X  [ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X:  Range = Actor[c3->s3] with generating sets:[128X[104X
    [4X[28X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][128X[104X
    [4X[28X  [ (5,7,6), (1,2)(3,4)(6,7) ][128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (1,3,2)(4,6,5) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (5,6,7), (1,2)(3,4)(6,7) ][128X[104X
    [4X[25Xgap>[125X [27XIsInjective( IMX3 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XZX3 := XModCentre( X3 ); [127X[104X
    [4X[28X[Group( () )->Group( () )][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
