Let $G$ be a group with identity element $e$. For $a,b \in G$, write $[a,b]=aba^{-1}b^{-1}$. Let $S=\{[a,b]:a,b\in G\}$, and let $[G,G]=\langle S\rangle$ denote the smallest subgroup of $G$ containing $S$. Then $[G,G]$ is normal in $G$; equivalently, for every $x\in G$ and every $h\in [G,G]$, one has $xhx^{-1}\in [G,G]$.