$N$ is a normal subgroup of $G$, when does $o(Na)=o(a)$

Question

If $N$ is a normal subgroup of $G$, I know that $o(Na)|o(a)$ for $a \in G/N$.

But in what cases does $o(Na)=o(a) $ ?

Answer 1

Let $e$ be the neutral element of $G$, $\langle a\rangle$ the subgroup generated by $a$. We have $$o(Na)=o(a)\Leftrightarrow \langle a\rangle \cap N=\{e\}. $$