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:
\(\ds X Y\) | \(=\) | \(\ds \paren {x N} \paren {y N}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x y N}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {y x N}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {y N} \paren {x N}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds Y X\) |
Thus $G / N$ is abelian.
$\blacksquare$