# Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?

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Closed subgroups of $\mathbf{Z}_p^n$ are $\mathbf{Z}_p$-submodules. So the theory of finitely generated modules over a PID works. You get a submodule of the free module of rank $n$, so this is free, i.e. isomorphic to $\mathbf{Z}_p^k$ for some $0\le k\le n$.
The theory of f.g. modules over PID actually gives a more precise result, namely describes how the module sits inside $\mathbf{Z}_p^n$. This is explained in many textbooks of undergraduate algebra.