# Quotient Group of Abelian Group is Abelian

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## 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.

$\blacksquare$