73.9 AutomorphismXMod

AutomorphismXMod( S [, A] )

This construction returns the crossed module whose range R is a permutation representation of a group A which is a group of automorphisms of the source S and which contains the inner automorphism group of S as a subgroup. When A is not specified the full automorphism group is used. The boundary morphism maps s in S to the representation of the inner automorphism of S by s. The action is the isomorphism R to A.

In the following example, recall that the automorphism group of the quaternion group is isomorphic to the symmetric group of degree 4 and that the inner automorphism group is isomorphic to k4. The group A is a subgroup of Aut(q8) isomorphic to d8.

    gap> q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );;
    gap> q8.name := "q8";; genq8 := q8.generators;;
    gap> iaq8 := InnerAutomorphismGroup( q8 );;
    gap> a := GroupHomomorphismByImages( q8, q8, genq8,
                  [(1,5,3,7)(2,6,4,8),(1,4,3,2)(5,6,7,8)]);;
    gap> genA := Concatenation( iaq8.generators, [a] );
    [ InnerAutomorphism( q8, (1,2,3,4)(5,8,7,6) ), 
      InnerAutomorphism( q8, (1,5,3,7)(2,6,4,8) ), 
      GroupHomomorphismByImages( q8, q8, [ (1,2,3,4)(5,8,7,6),
       (1,5,3,7)(2,6,4,8) ], [ (1,5,3,7)(2,6,4,8), (1,4,3,2)(5,6,7,8) ] ) ]
    gap> id := IdentityMapping( q8 );;
    gap> A := Group( genA, id );;
    gap> AX := AutomorphismXMod( q8, A );
    Crossed module [q8->PermSubAut(q8)] 
    gap> RecFields( AX );
    [ "isDomain", "isParent", "source", "range", "boundary", "action",
      "aut", "isXMod", "operations", "name", "isAutomorphismXMod" ]       

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GAP 3.4.4
April 1997