[GAP Forum] Automorphisms of abelian groups from presentation

[Dima Pasechnik]

Dear Danny,
On Tue, Sep 24, 2019 at 07:08:31AM -0400, Daniel Ruberman wrote:
> 
> Thanks for the response. To answer your question, both groups G and H are finite. The group H arises as a group of permutations of the basis elements of the free abelian groups V and W and is generated by transpositions of some of those basis elements.  

I see, this looks doable, e.g. via the usual finitely presented groups
technique of coset enumeration, assuming your |G| is "modest", say at
most 10^6  (even though 10^7 would still be doable).

Write down a faithful presentation for the extenion W.H, and enumerate
the cosets of V.H in W.H. The result will be a permutation group G.H,
(or perhaps not H, but a proper quotient of H)
with H the point stabilizer, and the orbits you are after will be
exactly the orbits of H in this action.
(also, the set G.H acts on will be G, with elements given as words in
generators of W).

I'd be happy to provide more details if needed.

Best,
Dima

> 
> Best,
> Danny
> 
> > On Sep 24, 2019, at 5:20 AM, Dima Pasechnik <dmitrii.pasechnik at cs.ox.ac.uk> wrote:
> > 
> > Dear Daniel,
> > On Sat, Sep 21, 2019 at 09:31:28PM -0400, Daniel Ruberman wrote:
> >> 
> >> 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. 
> > 
> > 
> > It might help to understand more about your problems: is your G finite?
> > If no, is your H finite? If no, is the action of H on G finite?
> > 
> > In the worst case, when you have infinitely many orbits of H on G,
> > things don't look doable in GAP (or in general, it looks like a hard
> > research problem to me).
> > 
> > 
> > Best
> > Dmitrii
> > http://users.ox.ac.uk/~coml0531/
>