Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)?

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.