Creation of Automorphism Groups

Automorphism groups are created using the AutomorphismGroup intrinsic functions. Currently these are limited to computing automorphism groups of soluble groups and permutation groups satisfying certain restrictions. For more details see below and the chapters on these groups. There is also a more basic function to construct a group of automorphisms of G by giving generators for G together with generator images under the action of the generating automorphisms being defined.

AutomorphismGroup(G): Grp -> GrpAuto

The function returns the automorphism group of G as a group of type GrpAuto. The algorithm used when G is a p-group of type GrpPC is that of Eick, Leedham-Green & O'Brien [ELGO02]. For more details see Section Automorphism Group Algorithm. When G has type GrpPC but is not a p-group, the algorithm of Smith [Smi94], as extended by Smith and Slattery, is used. For more details see Section Automorphism Group. When G is a permutation group, the non-abelian composition factors of G must lie in a restricted list. This list includes all simple groups of order at most 1.6times10^7, the alternating groups of degree at most 50 and the Mathieu groups. The algorithm used is that of Holt & Cannon [CH]. This relies on a database of automorphism groups for the simple factors of the given group, hence the restrictions referred to. See Section Automorphism Groups for more information.

AutomorphismGroup(G, Q, I): Grp, SeqEnum[GrpElt], SeqEnum[SeqEnum[GrpElt]] -> GrpAuto

Returns an automorphism group with base group G. The sequence Q of elements of G must generate G. Each element of the sequence I is a sequence of |Q| elements of G, which are the images of the given generators of G under some G-automorphism. An automorphism group is constructed with these |I| generators.