Suppose $G$ and $H$ are two countably infinite abelian groups such that every nontrivial element of $G \times H$ has order $7$. Then $G$ is isomorphic to $H$.
My idea is that each non-trivial element of $G$ and $H$ has order $7$. Therefore, we can consider them as vector spaces over ${\mathbb{Z}}/{7\mathbb{Z}}$. Since their order is the same, their dimension over $\mathbb{Z}/7{\mathbb{Z}}$ will also be the same. This implies that they are isomorphic as vector spaces and hence isomorphic as groups.
Is my approach correct? Is there any other way to prove this ?