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 <[email protected]> > > 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/ > _______________________________________________ Forum mailing list [email protected] https://mail.gap-system.org/mailman/listinfo/forum
