Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Commutator
Summary
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
[g, h] = g−1h−1gh.
This element is equal to the group's identity if and only if g and h commute (from the definition gh = hg [g, h] , being [g, h] equal to the identity if and only if gh = hg).
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
[g, h] = ghg−1h−1.
Commutator identities are an important tool in group theory. The expression ax denotes the conjugate of a by x, defined as x−1ax.
and
and
and
Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. 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.
Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:
If the derived subgroup is central, then
Rings often do not support division. Thus, the commutator of two elements a and b of a ring (or any associative algebra) is defined differently by
The commutator is zero if and only if a and b commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.