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" ]
GAP 3.4.4