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 $g^{n} \in H$?

No. Let $G=\mathbb{Z}\times\mathbb{Z}$, $H=\{(x,0)\}$. Then $(0,1)^n\notin H$ for any $n>0$
