Commutator subgroup

Let $G$ be a group . The commutator subgroup (or derived subgroup) of $G$ is $[G,G]=\langle [g,h] : g,h\in G\rangle,$ the subgroup generated by all commutators of elements of $G$.

The commutator subgroup is a normal subgroup of $G$. The quotient quotient group $G/[G,G]$ is abelian, and $[G,G]$ is the smallest normal subgroup $N\lhd G$ for which $G/N$ is abelian (equivalently: $G/[G,G]$ is the “largest” abelian quotient of $G$). Iterating commutator subgroups yields the derived series , central to solvability.

Examples:

• If $G$ is abelian, then $[G,G]=\{e\}$.
• In $S_3$, one has $[S_3,S_3]=A_3$ (the alternating subgroup of order $3$).
• In the dihedral group $D_8=\langle r,s\mid r^4=s^2=e,\ srs=r^{-1}\rangle$, the commutator subgroup is $\{e,r^2\}$.