Let $G$ be an abelian group, $S$ a subgroup. Let $H\cap S=0$ and $H$ is maximal among subgroups satisfying this relationship. Prove $G/(H+S)$ is torison.
This is not clear to me. If $S$ is a direct summand of $G$, then the question is trivial; otherwise in the finite generated case we would have $G=\prod \mathbb{Z}/p_{i}^{a_{i}}\mathbb{Z}$, and $S=\prod \mathbb{Z}/p_{i}^{b_{i}}\mathbb{Z},b_{i}\le a_{i}$. And $H$ cannot have any torsion elements. But I am not sure how to prove this in the general abelian group case.