Representations of an Automorphism Group

To compute with automorphism groups, Magma uses various concrete representations of the group. These are summarised in this section.

PermutationRepresentation(A) : GrpAuto -> Map, GrpPerm, SetIndx

ClassAction(A) : GrpAuto -> Map, GrpPerm, SetIndx

Compute a permutation representation of A. This routine finds a union of conjugacy classes of the base group G which is closed under the action of A and with G-normal closure equal to G. The permutation action of A on such a set is faithful. The results returned are the representation as a homomorphism A to P, the image of this homomorphism as a permutation group with standard support, and the set of elements of G used.

ClassImage(A) : GrpAuto -> GrpPerm

The image of the ClassAction permutation representation.

ClassUnion(A) : GrpAuto -> SetIndx

The set of base group elements used by the ClassAction function as its permutation domain.

FPGroup(A) : GrpAuto -> GrpFP, Map

A presentation for A on the generators of A. Also returns the isomorphism from the finitely presented group to A. If this is not already known it is computed using the ClassAction machinery.

OuterFPGroup(A) : GrpAuto -> GrpFP, Map

A finitely presented group O isomorphic to the outer automorphism group of the base group. Also returns the natural homomorphism from FPGroup(A) onto O.