# Commutator

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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