Given a finite abelian $p$-group and its factorization into groups of the form $\mathbb{Z}/p^k\mathbb{Z}$, does anyone know of a formula that gives the number of subgroups of a certain index/order? As I'm sure such a formula would contain some nasty product or sum, is there a computer algebra system out there that knows how to compute this?
Formula or code to compute number of subgroups of a certain order of an abelian $p$-group
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group-theory
computer-algebra-systems
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0I don't know if this will help, but [this question/answer] gives a way of describing all subgroups of a direct sum of cyclic groups of prime power order, in terms of certain congruences. – 2011-12-07
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0I probably need to code this up myself, since I've needed this a lot in the stuff that I've been working on lately. Somehow I think that needing this can't be too rare, so I would expect some CAS to have implemented this. :) – 2011-12-07
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0Check the `GAP` documentation, or ask in the `GAP` mailing list. – 2011-12-07
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0In answer to a similar question on MO, http://mathoverflow.net/questions/78956/a-question-on-the-number-of-subgroups-of-a-given-exponent-of-a-finite-abelian-p-g, Greg Martin gave a formula for the number of subgroups of a specific isomorphism type, so you could sum over the possible isomorphism types to get what you want. – 2011-12-07