On page 28 of A First Course in Modular Forms by Diamond and Shurman, the authors prove that every isogeny between complex tori is a composition of three isogenies: a multiplication by a natural number, a cyclic quotient isogeny, and an isomorphism.
There is one step in the proof that I do not understand. Namely, we have a subgroup $K$ of a group isomorphic to $\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$ and the authors say, "by the theory of finite Abelian groups $K\cong\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/nn'\mathbb{Z}$ for some positive integers $n$ and $n'$."
Why is this true? This looks like the theorem on submodules of a finitely-generated free module over a PID. Can we apply this result?
That is basically it. A subgroup of $(\Bbb Z/N\Bbb Z)^2$ pulls back to a subgroup $H$ of $\Bbb Z^2$ which contains $N\Bbb Z^2$. By the theory of the Smith Normal Form, we can take an automorphism of $\Bbb Z^2$ mapping $H$ to $H'=n\Bbb Z\times nn'\Bbb Z$. Then $H'\supseteq \Bbb Z^2$ so $nn'\mid N$.