Order of members of quotient groups

Question

$N$ is a normal subgroup of group finite group $G$. Let $a$ be an element of $G$, and $aN$ be an element of $G/N$ (quotient group). How to prove $O(aN) \mid O(a)$?

Answer 1

Hint: In any group, $O(x) \mid n$ if and only if $x^n = 1$.

Answer 2

More generally, if $\phi: G \to \Gamma$ is a homomorphism of groups, then $O(\phi(g))$ divides $O(g)$, simply because $g^n=1$ implies $\phi(g)^n=1$.

Apply this to the canonical projection $G \to G/N$.

Answer 3

Let $O(a) = n$, then take element $aN$ of quotient group, then $(aN)^n = N$ (identity in quotient group). If $m$ be order of $aN$, then $(aN)^m = N$, this must give $m \mid n$.