73.122 Actor crossed module

Actor( X )

The actor of a crossed module X is a crossed module Act(X) which has the Whitehead group (of regular derivations) as source group and the automorphism group PermAut(X) of X as range group. The boundary of Act(X) maps each derivation to the automorphism provided by XModEndomorphismDerivation. The action of an automorphism on a derivation is that provided by ImageAutomorphismDerivation.

    gap> ActXSC := Actor( XSC );
    Crossed module Actor[c3->s3]
    gap> XModPrint( ActXSC );

    Crossed module Actor[c3->s3] :- 
    : Source group WG([c3->s3]) has generators:
      [ (1,2,3)(4,6,5), (1,4)(2,5)(3,6) ]
    : Range group has parent ( PermAut(c3)xPermAut(s3) ) and has
        generators: [ (5,6,7), (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:
      (5,6,7) --> { source gens --> [ (1,2,3)(4,6,5), (1,6)(2,4)(3,5) ] }
      (1,2)(3,4)(6,7) -->
                  { source gens --> [ (1,3,2)(4,5,6), (1,4)(2,5)(3,6) ] }
      These 2 automorphisms generate the group of automorphisms.  

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