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Let $G$ be a finite abelian group of order $p^n$, where $p$ is a prime number. How to find the number of subgroups of order $p$?

i.e. find a formula for the number of subgroups of order $p$.

I know that $G$ is isomorphic to a direct product of cyclic $p$-groups. There are too many cases. I don't know the appropriate approach. It seems that there should be a formula that works universally.

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    How will your formula distinguish between, say, $\mathbf Z/4\mathbf Z$ and $\mathbf Z/2\mathbf Z \times \mathbf Z/2\mathbf Z$?2012-03-05
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    You could start by computing the number of subgroups of order $p$ in $\mathbb{Z}/p^k \mathbb{Z}$, and then use the fact $G$ is a product of such groups (from the classification of finite abelian groups)).2012-03-05

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