Order of Element in Quotient Group

Theorem
Let $G$ be a group, and let $H$ be a normal subgroup of $G$.

Let $G / H$ be the quotient group of $G$ by $H$.

The order of $a H \in G / H$ divides the order of $a \in G$.

Proof
Let $G$ be a group with normal subgroup $H$. $G/H$ is a homomorphic image of $G$, so we use $f:G \rightarrow G/H$ given by $f(a)=aH$ as our homomorphism. Suppose $a \in G$ and $a^n=e$ for some integer $n$. Then, by homomorphism,

$f \left(a^n\right)=f(a)^n=(aH)^n=a^{n}H^{n}=eH=H$, so $|H|$ divides $n$.