# 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

(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 **commutator subgroup** or **derived group** of *G* is the subgroup generated by all commutators, written or . It 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.

Commutators of higher order are defined iteratively as

The higher derived groups are defined as , and so on.

## Ring theory

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