# Quotient Group of Abelian Group is Abelian

## Theorem

Let $G$ be an abelian group.

Let $N \le G$.

Then the quotient group $G / N$ is abelian.

## Proof

First we note that because $G$ is abelian, from Subgroup of Abelian Group is Normal we have $N \lhd G$.

Thus $G / N$ exists for all subgroups of $G$.

Let $X = x N, Y = y N$ where $x, y \in G$.

From the definition of coset product:

 $\displaystyle X Y$ $=$ $\displaystyle \paren {x N} \paren {y N}$ $\displaystyle$ $=$ $\displaystyle \paren {x y N}$ $\displaystyle$ $=$ $\displaystyle \paren {y x N}$ $\displaystyle$ $=$ $\displaystyle \paren {y N} \paren {x N}$ $\displaystyle$ $=$ $\displaystyle Y X$

Thus $G / N$ is abelian.

$\blacksquare$