Anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
The commutator of two elements, g and h, of a group G, is the element
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 G 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.
The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as
Commutator identities are an important tool in group theory. The expression ax denotes the conjugate of a by x, defined as x−1a x.
Identity 5 is also known as the Hall–Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).
N.B. The above definition of the conjugate of a by x is used by some group theorists. Many other group theorists define the conjugate of a by x as xax−1. This is often written . Similar identities hold for these conventions.
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