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Suppose I have a group $G$ with a finite presentation, and I know that $G$ is polycyclic. So I input the group into GAP via generators and relators. Now I want to find a presentation of an infinite index subgroup; I can do this by hand (usually), by doing a Reidemeister-Schreier rewriting, and then simplifying the presentation as best I can. GAP will, in general, not attempt to find presentations of infinite index subgroups. But if I tell GAP the group is polycyclic, are there various algorithms out there such that GAP can in fact give me a finite presentation for this infinite index subgroup?

If that is too much, can GAP do it for certain "nice" subgroups (the commutator, etc.)?

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    @Steve: Bettina Eick has not released the source code. CallPCQA is probably Eddie Lo's old (1990-1995ish) pcqa. Like all pcqa known to me it has some problems, but more recent pcqa are usually better. Bettina indicated Max Horn would be happy to test his new code on any presentations you wanted to send him.2011-03-14

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To do what you want, you need a tool that takes a finitely presented group, and, given the information that this group is polycyclic, determines a polycyclic presentation for this group. In general, this is a computationally very difficult problem, although in certain special cases it may be much simpler.

The best approach that I know to this problem in general is to apply a polycyclic quotient algorithm, such as the one described in the paper "A Polycyclic Quotient Algorithm" by Eddie Lo (J. Symbolic Computation (1998) 25, 61–97). He implemented this in C and GAP3, which makes it a tad difficult to use on modern day computers.

The idea then is to take your finite presentation, and compute a maximal PC quotient of it. Since you know your group is polycyclic, this will definitely terminate, but it might take a looong time and a lot of memory (and loong could easily mean more than your life time if things are bad... :/).

But once you have a pc presentation, you can usually work quite effectively with it, with the help of the polycyclic package, as Jack Schmidt already pointed out.

Anyway, if you still want to try to find a pc presentation for a specific group, I implemented a pc quotient algorithm in GAP 4 last year, which can perhaps solve your problem. It's not yet been published, though I hope to get this finally done sometime soon. In the meantime, feel free to get in touch with my privately and perhaps I can help you in concrete cases.