Commutator
From Citizendium, the Citizens' Compendium
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
![[x,y] = x^{-1} y^{-1} x y \,](/images/math/f/5/5/f550df607d5def06feee9bc1fe4bc057.png)
(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 ![[G,G]](/images/math/3/9/7/3977a81444c9d8799fad00e62a3b3f78.png)
. 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,\ldots,x_{n-1},x_n] = [x_1,[x_2,\ldots,[x_{n-1},x_n]\ldots]] . \,](/images/math/7/f/4/7f473480f5d3054e15f9c79a40d15a26.png)
The higher derived groups are defined as ![G^{(1)} = [G,G]](/images/math/e/c/c/eccc447c06e372bd40f22ec942f4d33c.png)
, ![G^{(2)} = [G^{(1)},G^{(1)}]](/images/math/1/5/f/15ffd5fd9cffbf8c2ed3f2ef83a03da4.png)
and so on.
Ring theory
In a ring, the commutator of elements x and y may be defined as
![[x,y] = x y - y x . \,](/images/math/8/b/6/8b644d8995387703cf750813b7ccb3b0.png)
