For $G$ an abelian group and $H$ a subgroup, is $[G : H]$ the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Question

Let $G$ be an abelian group and $H$ a subgroup.

What is the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Is it $[G : H]$, or can it be strictly smaller (a divisor of $[G : H]$)?

Answer 1

This already fails when $H$ is trivial group. Consider $G=\mathbb{Z}/p \oplus \mathbb{Z}/p$ for example.