Classifying subgroups of $\mathbb{Z}^n$ up to isomorphism

Question

I believe that the only subgroups of $\mathbb{Z}^n$ up to isormorphism are $\{0\}$ and $\mathbb{Z}^m$, with $m\leq n$.

This because if $z\neq 0\hookrightarrow H<\mathbb{Z}^n$ so $\langle z\rangle\sim \mathbb{Z}\hookrightarrow H$.

However, I'd like to get this formally (for instance, by a result on a lemma or a more formal proof). Could you help me?

Thank you so much

Answer 1

Since $\Bbb Z^n$ is a finitely-generated abelian group, the result follows from the Fundamental Theorem of Finitely-generated Abelian Groups, which is a classification result described in this Wikipedia article.