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 Failed to parse (Missing texvc executable; please see math/README to configure.): G
N.B. The above definition of the commutator is used by some group theorists. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): {}^x a . Similar identities hold for these conventions.
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