Commutator

Let $G$ be a group and let $g,h\in G$. The commutator of $g$ and $h$ is the element $[g,h]=g^{-1}h^{-1}gh.$ (Warning: some authors use the inverse convention $ghg^{-1}h^{-1}$; this changes commutators by inversion, but does not change the subgroup they generate.)

The commutator satisfies $[g,h]=e$ if and only if $gh=hg$. In particular, if $g$ lies in the center of $G$, then $[g,h]=e$ for all $h\in G$. The subgroup generated by all commutators is the commutator subgroup ; a group is abelian if and only if all commutators are trivial.

Examples:

• In any abelian group, $[g,h]=e$ for all $g,h$.
• In $S_3$ (with $[g,h]=g^{-1}h^{-1}gh$), if $g=(12)$ and $h=(23)$, then $[g,h]=(132)\neq e$, so $g$ and $h$ do not commute.
• If $g=h$, then $[g,g]=e$ in every group.