Quotient Group is Abelian iff All Commutators in Divisor

Theorem
Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

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


 * $\forall x, y \in G: \sqbrk {x, y} \in N$

where $\sqbrk {x, y}$ denotes the commutator of $x$ and $y$.

That is, $x y x^{-1} y^{-1} \in N$ for all $x, y \in G$.