If $g$ is an element in an abelian group $G$ and $H\leqslant G$, must there exist an $n$ such that $g^n\in H$? Let $G$ be an abelian group and $H$ a subgroup of $G$. For each $g \in G$, does there always exist an integer $n$ such that $g^{n} \in H$?

Hint: For $n\geq 1$: we can find find a counterexample subgroup $H\leq G$, with $G =(\mathbb{Z}, +)$ which is infinite and cyclic: Put $H = \{e\} = \{0\} \leq G$.
