[GAP Forum] Automorphisms of abelian groups from presentation

[Daniel Ruberman]

Dear all,

I would like to do a computation of the following sort. I have a finitely presented Abelian group G, presented by an exact sequence V \to W \to G \to 0, where each of V and W is free abelian. I have an explicit matrix A for the first map, and hence (using Smith normal form) generators for G. My problem is rather symmetric, so there is a large group H that acts on V and W, commuting with the matrix A. Hence H becomes a group of automorphisms of G.

I would like to know (representatives for) the orbits of H on G, written in terms of this presentation. In other words I would like representatives for the orbits, written as elements of W. I know that GAP can find orbits of group actions of many sorts, but I don’t see how to implement this in my situation. 

Thanks in advance for any help.

Daniel Ruberman