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