Quotient Group of Abelian Group is Abelian

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$