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I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.

Are there simple formulas if one restricts to low rank for the subgroups? For example, are there formulas for enumerating cyclic subgroups, or subgroups whose minimal number of generators is $2$?

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up vote 7 down vote accepted

I think that you can find the formulas that you are looking for in the paper "An arithmetic method of counting the subgroups of a finite abelian group" by Marius Tarnauceanu, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 4, 373–386.

In particular, Theorem 4.3 seems relevant, but there are other results that might be of interest to your question.

The paper can be downloaded from the journal's website.

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Thank you very much for your answer! I am very surprised that the answer to this question seems to be the main topic of a contemporary research paper. I would have expected that at least the cyclic case would be reasonably doable. –  user22518 Apr 4 '12 at 12:57
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