Commutator

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.
Group theory.
The commutator of two elements, and , of a group , is the element
This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ).
The set of all commutators of a group is not in general closed under the group operation, but the subgroup of "G" generated by all commutators is closed and is called the "derived group" or the "commutator subgroup" of "G". Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group.
The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as
Identities (group theory).
Commutator identities are an important tool in group theory. The expression denotes the conjugate of by , defined as .
Relation (3) is called anticommutativity, while (4) is the Jacobi identity.
Additional identities.
Some of the above identities can be extended to the anticommutator using the above ± subscript notation.
For example:
Exponential identities.
Consider a ring or algebra in which the [[exponential function|exponential]] formula_25 can be meaningfully defined, such as a [[Banach algebra]] or a ring of [[formal power series]].
In such a ring, [[Hadamard's lemma]] applied to nested commutators gives: formula_26 (For the last expression, see "Adjoint derivation" below.) This formula underlies the [[Baker–Campbell–Hausdorff formula#An important lemma|Baker–Campbell–Hausdorff expansion]] of log(exp("A") exp("B")).
A similar expansion expresses the group commutator of expressions formula_27 (analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets),
Graded rings and algebras.
When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as 
Adjoint derivation.
Especially if one deals with multiple commutators in a ring "R", another notation turns out to be useful. For an element formula_30, we define the adjoint mapping formula_31 by:
This mapping is a derivation on the ring "R": 
By the Jacobi identity, it is also a derivation over the commutation operation: 
Composing such mappings, we get for example formula_35 and formula_36 We may consider formula_37 itself as a mapping, formula_38, where formula_39 is the ring of mappings from "R" to itself with composition as the multiplication operation. Then formula_37 is a Lie algebra homomorphism, preserving the commutator:
By contrast, it is not always a ring homomorphism: usually formula_42.
General Leibniz rule.
The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:
Replacing "x" by the differentiation operator formula_44, and "y" by the multiplication operator formula_45, we get formula_46, and applying both sides to a function "g", the identity becomes the usual Leibniz rule for the "n"-th derivative formula_47.