Find non-trivial groups $G, H$ such that $G \cong H \times G$

Question

This is question II.3.4 from Aluffi's algebra 0.

Let $G, H$ be groups and assume that $G \cong H \times G$. Can you conclude that $H$ is trivial?

The hint already states this is not the case. I think the group should be infinite. So, I have tried doing several things with the real numbers both additive and multiplicative and a small group such as $\{1, -1\}$, but I have not been able to find a bijective homomorphism in this way.

Answer 1

Let $H$ be the additive group of the field $\mathbb F_2$ (i.e. the only group up to isomorphism with 2 elements) and $G = \oplus_{i-1}^\infty H$ be the infinite product group composed of countable copies of $H$.

Then $G \cong H \times G$.