A counterexample to Cantor–Schröder–Bernstein in groups.

Question

We need to find a counterexample to the Cantor–Schröder–Bernstein theorem for the category of groups (that is, find two monomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to G$ such that $G$ and $H$ are non isomorphic). The professor suggested us to consider the groups $G = \bigoplus_{i \geq 1} \mathbb{Z}_{2^i}$ and $H = \bigoplus_{i \geq 2} \mathbb{Z}_{2^i}$. I constructed the following monomorphisms:

Both are indeed monomorphisms, but I haven’t been able to prove that $G$ and $H$ are non-isomorphic. Does anybody have an idea? May I get a hint?

Answer 1

Every element of $H$ of order $2$ is twice some element.

But $G$ has elements of order $2$ which don't have that property (e.g., $(1,0,0,0,...))$.