  
  [1X6 [33X[0;0YActors of 2d-groups[133X[101X
  
  
  [1X6.1 [33X[0;0YActor of a crossed module[133X[101X
  
  [33X[0;0YThe  [13Xactor[113X  of  [22XmathcalX[122X  is  a  crossed  module  [22X(∆ : mathcalW(mathcalX) ->
  Aut(mathcalX))[122X  which was shown by Lue and Norrie, in [Nor87] and [Nor90] to
  give  the  automorphism  object  of  a  crossed  module  [22XmathcalX[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 of the actor is a
  permutation  representation  [22XA[122X  of  the  automorphism group [22XAut(mathcalX)[122X of
  [22XmathcalX[122X.[133X
  
  [1X6.1-1 AutomorphismPermGroup[101X
  
  [29X[2XAutomorphismPermGroup[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XGeneratingAutomorphisms[102X( [3Xxmod[103X ) [32X attribute
  [29X[2XPermAutomorphismAsXModMorphism[102X( [3Xxmod[103X, [3Xperm[103X ) [32X attribute
  
  [33X[0;0YThe automorphisms [22X( σ, ρ )[122X of [22XmathcalX[122X form a group [22XAut(mathcalX)[122X of crossed
  module  isomorphisms.  The  function  [10XAutomorphismPermGroup[110X  finds  a set of
  [10XGeneratingAutomorphisms[110X for [22XAut(mathcalX)[122X, and then constructs a permutation
  representation  of  this  group,  which  is  used  as the range of the actor
  crossed  module of [22XmathcalX[122X. The individual automorphisms can be constructed
  from       the      permutation      group      using      the      function
  [10XPermAutomorphismAsXModMorphism[110X.  The  example  below uses the crossed module
  [10XX3=[c3->s3][110X constructed in section [14X5.1[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[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 [27XSize( APX3 );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XgenX3 := GeneratingAutomorphisms( X3 );    [127X[104X
    [4X[28X[ [[c3->s3] => [c3->s3]], [[c3->s3] => [c3->s3]] ][128X[104X
    [4X[25Xgap>[125X [27Xe6 := Elements( APX3 )[6];[127X[104X
    [4X[28X(1,2)(3,4)(5,7)[128X[104X
    [4X[25Xgap>[125X [27Xm6 := PermAutomorphismAsXModMorphism( X3, e6 );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( m6 );[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 = Source[128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (1,3,2)(4,5,6) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (4,6,5), (2,3)(4,5) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 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 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[22XmathcalX = (∂ : S -> R),~[122X the initial crossed module, on the left,[133X
  
  [30X    [33X[0;6Y[22XmathcalW(mathcalX)  =  (η  : S -> W),~[122X the Whitehead crossed module of
        [22XmathcalX[122X, at the top,[133X
  
  [30X    [33X[0;6Y[22XmathcalN(X) = (α : R -> A),~[122X the Norrie crossed module of [22XmathcalX[122X, at
        the bottom,[133X
  
  [30X    [33X[0;6Y[22XAct(mathcalX)  = ( ∆ : W -> A),~[122X the actor crossed module of [22XmathcalX[122X,
        on the right, and[133X
  
  [30X    [33X[0;6Y[22XmathcalL(mathcalX)  = (∆∘η = α∘∂ : S -> A),~[122X the Lue crossed module of
        [22XmathcalX[122X, along the top-left to bottom-right diagonal.[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 [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  [ (1,2,3)(4,5,6) ][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 );; [127X[104X
    [4X[25Xgap>[125X [27XDisplay( 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  [ (5,7,6), (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), (1,6)(2,5)(3,4) ] }[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[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  [ (1,2,3)(4,5,6) ][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 --> [ (1,2,3)(4,5,6) ] }[128X[104X
    [4X[28X  (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,6,5) ] }[128X[104X
    [4X[28X  These 2 automorphisms generate the group of automorphisms.[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-3 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,
  the  inner  morphism  of  [10XX3=(c3->s3)[110X,  from  Chapter  [14X5[114X, is an inclusion of
  crossed modules.[133X
  
  [33X[0;0YNote  that  we  appear  to  have  defined  [13Xtwo[113X sorts of [13Xcentre[113X for a crossed
  module:   [10XXModCentre[110X   here,  and  [2XCentreXMod[102X  ([14X4.1-7[114X)  in  the  chapter  on
  isoclinism.  We suspect that these two definitions give the same answer, but
  this remains to be resolved.[133X
  
  [4X[32X  Example  [32X[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  [ (1,2,3)(4,5,6) ][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
  
