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