Mar 9, 2017

Normality of the Commutator Subgroup

The problem is:
Given a group $G$, show that a subgroup $N = \{xyx^{-1}y^{-1} \mid x,y \in G \}$ is normal.
Since it was difficult for me, I searched online and found a proof on Stack Exchange. As many others, I think it is slick.

Let $g \in G$ and $n,n' \in N$. Then, $$ gng^{-1} = gng^{-1}(n^{-1}n) = (gng^{-1}n^{-1})n = n'n \in N .$$