Commutator

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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

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

[ x , y ] = x − 1 y − 1 x y {\displaystyle [x,y]=x^{-1}y^{-1}xy\,}

(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  G ( 1 ) {\displaystyle G^{(1)}} or  [ G , G ] {\displaystyle [G,G]} . 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

[ x 1 , x 2 , … , x n − 1 , x n ] = [ x 1 , [ x 2 , … , [ x n − 1 , x n ] … ] ] . {\displaystyle [x_{1},x_{2},\ldots ,x_{n-1},x_{n}]=[x_{1},[x_{2},\ldots ,[x_{n-1},x_{n}]\ldots ]].\,}

The higher derived groups are defined as  G ( 1 ) = [ G , G ] {\displaystyle G^{(1)}=[G,G]} ,  G ( 2 ) = [ G ( 1 ) , G ( 1 ) ] {\displaystyle G^{(2)}=[G^{(1)},G^{(1)}]} and so on.

## Ring theory[edit]

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

[ x , y ] = x y − y x . {\displaystyle [x,y]=xy-yx.\,}