## Two cosets commute precisely when the subgroup contains the commutator of two representatives

Let $G$ be a group and $N \leq G$ a normal subgroup. Prove that $x{}N$ and $yN$ commute in $G/N$ if and only if $x^{-1}y^{-1}xy \in N$. (The element $[x,y] = x^{-1}y^{-1}xy$ is called the commutator of $x$ and $y$ in $G$.)

We have $(xN)(yN) = (yN)(xN)$ if and only if $(xy)N = (yx)N$, if and only if $(yx)^{-1}(xy) = x^{-1}y^{-1}xy \in N$.