Commutator

In algebra, the commutator of two elements of an algebraic structure is a measure of whether the algebraic operation is commutative.

Group theory
In a group, written multiplicatively, the commutator of elements x and y may be defined as


 * $$ [x,y] = x^{-1} y^{-1} x y \, $$

(although variants on this definition are possible). Elements x and y commute if and only if the commutator [x,y] is equal to the group identity. The subgroup of G generated by all commutators, written [G,G], is normal and indeed characteristic and the quotient G/[G,G] is abelian. A quotient of G by a normal subgroup N is abelian if and only if N contains the commutator subgroup.

Ring theory
In a ring, the commutator of elements x and y may be defined as


 * $$ [x,y] = x y - y x . \, $$