Problem: Let $G$ be an infinite abelian group. Show that if $G$ has a nontrivial subgroup $K$ such that $K$ is contained in all nontrivial subgroups of $G$, then $G$ is a $p$- group for some prime $p$. Moreover, $G$ is of type $p^\infty$ (quasicyclic) group.
I have the following result:
If $G$ is an infinite abelian group all of whose proper subgroup are finite, then $G$ is of type $p^\infty$ group for some prime $p$.
So I am trying to show that each of subgroup of $G$ is finite. But I can't see anymore. Please help me to find some direction.