This seemingly simple question has puzzled me for a while: Determine which abelian groups A fit into a short exact sequence $$0 \to \mathbb{Z}_{p^m} \to A \to \mathbb{Z}_{p^n} \to 0$$
(This is question 2.1.14 in Hatcher's Algebraic Topology)
First and foremost, I'd like to see an elementary straightforward solution. If there is a more sophisticated solution, I'd be happy to see it as well (one approach we had in mind is to compute the value of Ext, which didn't work quite well).