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[edit]

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

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

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

${\displaystyle [x_{1},x_{2},\ldots ,x_{n-1},x_{n}]=[x_{1},[x_{2},\ldots ,[x_{n-1},x_{n}]\ldots ]].\,}$

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

Ring theory[edit]

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

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