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 All Subgroups of Abelian Group are Normal we have $$N \triangleleft G$$.

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 Product of Cosets:

$$ $$ $$ $$ $$

Thus $$G / N$$ is abelian.