Equality between side classes with a finite and infinite cyclic quotient group

Question

Let $G/H$ be a cyclic group and $a^kH \in G/H,$ with $k \in \mathbb{Z}$.

I have a small doubt: if for some reason $a^k \in H,$ then $k = 0$ if $G/H$ is an infinite cyclic group? What if $|G/H| = n,$ then if $a^k \in H,$ that means $n \mid k?$

Answer 1

By closure of $H$ (as a subgroup of $G$), if $a\in H$, then $a^m\in H$ for all $m\in\Bbb Z$.