Commutator

The commutator of two elements, g and h, of a group G, is the element

= g−1h−1gh.

It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.

N.B. The above definition of the commutator is used by some group theorists. Many other group theorists define the commutator as

= ghg−1h−1.