In a group G, an element [x, y]=x−1 y−1 xy for x, y in G. Note this is always the identity if G is abelian. The commutators generate a normal subgroup known as the commutator subgroup, or derived subgroup, denoted [G,G]. The quotient group G/[G,G] is abelian and called the abelianization of G. See also derived series.
In a ring, elements x and y have commutator [x,y] = xy − yx. Such commutators are important in quantum theory, with Heisenberg’s uncertainty principle being an inequality involving a commutator of two observables.