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
![{\displaystyle [x,y]=x^{-1}y^{-1}xy\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/669e32fa7cb3d478ce19bc24b0f500519d5372e5)
(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.
![{\displaystyle [G,G]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ddf7a724a331d1e12ffa6571ba246ebf08f1335)
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 ]].\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e01c002daf8c9b80c5d297be9594a7abf8cf697)
The higher derived groups are defined as , and so on.
![{\displaystyle G^{(1)}=[G,G]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214618d2144d71452dc80051b136033a49b88e9b)
![{\displaystyle G^{(2)}=[G^{(1)},G^{(1)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67f1e8a1883662ecbb8f679ed3f03274d9f4e334)
Ring theory
In a ring, the commutator of elements x and y may be defined as
![{\displaystyle [x,y]=xy-yx.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7128822399d11b7613dd124ef3cd7ecd7824c1d9)
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 123-124.