Is it true that an abelian group is cyclic iff every $n$-torsion subgroup is cyclic?

Question

Obviously if it is cyclic every subgroup, including the torsion subgroups, is cyclic. I'm having trouble figuring if the converse is true.

Answer 1

No. Consider the circle group $\{e^{i\theta}\mid\theta\in [0,2\pi)\}$.

For the finite case, it's true. For, if $G$ is not cyclic, it contains a subgroup isomorphic to $\Bbb Z_p×\Bbb Z_p$. But then, the $p$-torsion subgroup is not cyclic.