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.

Examples of characteristic subgroups include the commutator subgroup and the center of a group.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

In other words, G / N is abelian if and only if N contains the commutator subgroup.

So in some sense it provides a measure of how far the group is from being abelian ; the larger the commutator subgroup is, the " less abelian " the group is.

This motivates the definition of the commutator subgroup ( also called the derived subgroup, and denoted G ′ or G < sup >( 1 )</ sup >) of G: it is the subgroup generated by all the commutators.

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below.

Moreover, taking G = H it shows that the commutator subgroup is stable under every endomorphism of G: that is, is a fully characteristic subgroup of G, a property which is considerably stronger than normality.

The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g

* Metabelian group is a group where the commutator subgroup is abelian

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections.

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

The above definition of the commutator is used by some group theorists.

It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator ( see next section ).

The commutator of two elements a and b of a ring or an associative algebra is defined by

The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously.

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

For elements g and h of a group G, the commutator of g and h is.

The commutator is equal to the identity element e if and only if gh

An element of G which is of the form for some g and h is called a commutator.

The identity element e = is always a commutator, and it is the only commutator if and only if G is abelian.

Affine Lie algebras play an important role in string theory and conformal field theory due to the way they are constructed: starting from a simple Lie algebra, one considers the loop algebra,, formed by the-valued functions on a circle ( interpreted as the closed string ) with pointwise commutator.

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers ( because R is the Lie algebra of the Lie group of positive real numbers with multiplication ), for complex numbers ( because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication ) and for matrices ( because M < sub > n </ sub >( R ) with the regular commutator is the Lie algebra of the Lie group GL < sub > n </ sub >( R ) of all invertible matrices ).

This was necessary because a continuously moving power switch known as the commutator is needed to keep the field correctly aligned across the spinning rotor.

Obviously, commutator relations are quite different than in the Schrödinger picture because of the time dependency of operators.

Brown, using the experience he had gained while working for Jean Heilmann on steam-electric locomotive designs, had observed that three-phase motors had a higher power-to-weight ratio than DC motors and, because of the absence of a commutator, were simpler to manufacture and maintain.

* The symmetric group S < sub > 4 </ sub > of order 24 is solvable but is not metabelian because its commutator subgroup is the alternating group A < sub > 4 </ sub > which is not abelian.

However commutator motors are noisier than induction motors, partially due to the higher speeds and partially because the commutator brushes rub on the slotted armature.

Informally we can think of elements of the Lie algebra as elements of the group that are " infinitesimally close " to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements.

A DC motor is powered by direct current, although there is almost always an internal mechanism ( such as a commutator ) converting DC to AC for part of the motor.

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices ( or endomorphisms of a vector space ) in such a way that the Lie bracket is given by the commutator.

If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

To find the conditions on the constants u and v such that the transformation remains canonical, the commutator is expanded, viz.

The Fialka design seems to derive from the Swiss NEMA, but the NEMA only has 5 electrical rotors vs. the Fialka's 10 and NEMA lacks a punched card commutator or an equivalent, such as a plug board.

such that E = E, where Z ( E ) denotes the center of E and denotes the commutator.

Since at low frequencies, such as the AM broadcast band, the near field region is physically quite large, this provides a considerable benefit in relation to static generating devices ( such as sparking at the commutator of an electric motor ) in the vicinity.

In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients ( equivalently, its abelianization, which is the universal abelian quotient, is trivial ).

Equivalently, a group is quasisimple if it is isomorphic to its commutator subgroup and its inner automorphism group Inn ( G ) ( its quotient by its center ) is simple ; due to Grün's lemma, Inn ( G ) must be non-abelian.

Many other group theorists define the commutator as

Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket ( algebra ) commutators,

Here are some simple but useful commutator identities, true for any elements s, g, h of a group G:

It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96 ; in fact there are two nonisomorphic groups of order 96 with this property.

The generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra.