Commutator of two elements

Definition

Let $G$ be a group and let $g, h \in G$. The commutator of $g$ and $h$ is the element $[g, h] = g^{-1} h^{-1} g h$ of $G$.

Remarks

The equation $gh = hg [g, h]$ holds, so $[g, h] = e$ if and only if $gh = hg$. Some authors use the convention $[g, h] = g h g^{-1} h^{-1}$ instead. The inverse of $[g, h]$ is $[h, g]$.

Explanations

L1 human

The commutator measures how far two elements are from commuting. It is the correction factor that you need when you swap $g$ and $h$ in a product. In an abelian group, every commutator is the identity.