73.8 CentralExtensionXMod

CentralExtensionXMod( f )

This construction returns the crossed module whose boundary f is a surjection from S to R having as kernel a subgroup of the centre of S. The action maps an element of r in R to conjugation of S by f^{-1}r.

    gap> d8 := Subgroup( s4, [ (1,2,3,4), (1,3) ] );; d8.name := "d8";;
    gap> gend8 := d8.generators;; genk4 := k4.generators;;
    gap> f := GroupHomomorphismByImages( d8, k4, gend8, genk4 );;
    gap> EX := CentralExtensionXMod( f );
    Crossed module [d8->v4] 
    gap> XModPrint( EX );
    Crossed module [d8->v4] :-
    : Source group d8 has parent s4 and generators:
      [ (1,2,3,4), (1,3) ]
    : Range group k4 has parent s4 and generators:
      [ (1,2)(3,4), (1,3)(2,4) ]
    : Boundary homomorphism maps source generators to:
      [ (1,2)(3,4), (1,3)(2,4) ]
    : Action homomorphism maps range generators to automorphisms:
      (1,2)(3,4) --> { source gens --> [ (1,2,3,4), (2,4) ] }
      (1,3)(2,4) --> { source gens --> [ (1,4,3,2), (1,3) ] }
    These 2 automorphisms generate the group of automorphisms.   

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GAP 3.4.4
April 1997