Commutator subgroup

The commutator subgroup $G^{\prime }$ is generated by all $[x,y]=x^{-1}{y}^{-1}xy$. It is characteristic, $G/G^{\prime }$ is abelian, and every normal subgroup with abelian quotient contains $G^{\prime }$.

A group $G$ is perfect when $G^{\prime }=G$, equivalently when its abelianization is trivial.

The abelianization of a group is the quotient $G^{\mathrm{ab}}=G/G^{\prime }$ by its commutator subgroup. It is the largest abelian quotient of $G$.

For odd $n\ge 3$, the dihedral group $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The group commutator $[r,s]=r^{-2}$ generates the rotation subgroup, while the quotient by that subgroup is $C_2$. Hence the commutator subgroup is exactly $⟨r⟩$ and the abelianization is $C_2$. The only abelian quotients are $C_2$ and the trivial group.