Take the 2-minute tour ×
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.

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|improve this question
add comment

1 Answer

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|improve this answer
add comment

Your Answer

 
discard

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.