Tessellate S.T.E.M.S - Mathematics - College - Set 2 - Problem 4

Let $G$ and $H$ be non-isomorphic abelian groups, $i : G \rightarrow H$ be an injective group homomorphism and $j : H \rightarrow G$ be an injective group homomorphism.

$(a)$ $G$ and $H$ can both be finite.

$(b)$ $G$ and $H$ can both be finitely generated.

$(c)$ $G$ and $H$ can both be countably generated.

$(d)$ None of the above

$\textbf{Hint}$ - You may require Stucture theorem for finitely generated abelian groups.

---

This problem is a part of Tessellate S.T.E.M.S.