Let G be a group. Prove that $\displaystyle N = \langle x^{-1} y^{-1} xy \, \, | \, \, x, y \in G \rangle$ is a normal subgroup of G and $\displaystyle G/N$ is abelian

So starting with the first part...

To show something is a normal subgroup we have to show that $\displaystyle g (x^{-1} y^{-1} xy) g^{-1} \in N$ for some g in G.

I've been trying to rearrange that equation to look something line the definition of N but can't seem to get anywhere...

$\displaystyle g x^{-1} y^{-1} xy g^{-1} \cong g^{-1} x^{-1} y^{-1} xy g = (yxg)^{-1} xyg = g^{-1} (yx)^{-1} (xy)g = ...$

??