# 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 $H$.

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

From Quotient Group Epimorphism is Epimorphism, $G / H$ is a homomorphic image of $G$.

Let $q_H: G \to G / H$ given by $\map f a = a H$ be that quotient epimorphism.

Let $a \in G$ such that $a^n = e$ for some integer $n$.

Then, by the morphism property of $q_H$:

 $\ds \map {q_H} {a^n}$ $=$ $\ds \paren {\map {q_H} a}^n$ $\ds$ $=$ $\ds \paren {a H}^n$ $\ds$ $=$ $\ds a^n H$ Definition of Coset Product $\ds$ $=$ $\ds e H$ by hypothesis $\ds$ $=$ $\ds H$

Hence $\order H$ divides $n$.

$\blacksquare$