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Wikipedia / Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G of elements of G that commute with every element of S such that conjugation leaves each element of S fixed. The normalizer of S in G is the set of G fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsetsS.

Suitably formulated, the definitions also apply to semigroups.

In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

Definitions

Group and semigroup

The centralizer of a subset of group (or semigroup) G is defined as

where only the first definition applies to semigroups. If there is no ambiguity about the group in question, the G is a singleton set, we write CG(a) instead of CG({a}). Another less common notation for the centralizer is Z(a), which parallels the notation for the center. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, Z(g).

The normalizer of S in the group (or semigroup) G is defined as

is a normal subgroup. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of and s is in , then it must be that , but if g is in the normalizer, then for some t in , with t possibly different from s. That is, elements of the centralizer of must commute pointwise with , but elements of the normalizer of S need only commute with S as a set. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure.

.

Ring, algebra over a field, Lie ring, and Lie algebra

If R is a ring or an algebra over a field, and is a subset of R, then the centralizer of is exactly as defined for groups, with R in the place of G.

is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset is defined to be

The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If R is an associative ring, then R can be given the bracket product . Of course then if and only if . If we denote the set R with the bracket product as LR, then clearly the ring centralizer of in R is equal to the Lie ring centralizer of in LR.

The Lie bracket can also be viewed as an operation of the set on itself, because . The Lie bracket makes Thus, the centralizer is defined in the same way for Lie algebras as for groups.

The normalizer of a subset is given by

While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set . If is the largest Lie subring (or Lie subalgebra, as the case may be) in which is a Lie ideal.

Example

Consider the group

  1. (the symmetric group of permutations of 3 elements).

:

is the permutation that fixes the first element and swaps the second and third element.

:

  1. .

  2. .

  3. .

  4. .

  5. .

  6. .

under conjugation.

. It's clear in this example that the only such element in S3 itself ([1, 2, 3], [1, 3, 2]).

Properties

Semigroups

forms a subsemigroup; i.e. a commutant is its own bicommutant.

Groups

Source:

  • The centralizer and normalizer of are both subgroups of G.

  • Clearly, . In fact, CG(S) is always a normal subgroup of NG(S), being the kernel of the homomorphism and the group NG(S)/CG(S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as , and especially if the torus is maximal (i.e. it is a central tool in the theory of Lie groups.

  • CG(CG(S)) contains , but CG(S) need not contain . Containment occurs exactly when is abelian.

  • If H is a subgroup of G, then NG(H) contains H.

  • If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup NG(H).

  • If is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains is the subgroup CG(S).

  • A subgroup H of a group G is called a of G if .

  • The center of G is exactly CG(G) and G is an abelian group if and only if .

  • For singleton sets, .

  • By symmetry, if and T are two subsets of G, if and only if .

  • For a subgroup H of group G, the N/C theorem states that the factor group NG(H)/CG(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H. Since and , the N/C theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G.

  • If we define a group homomorphism by , then we can describe NG(S) and CG(S) in terms of the group action of Inn(G) on G: the stabilizer of in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing pointwise is T(CG(S)).

  • A subgroup H of a group G is said to be C-closed or self-bicommutant if for some subset . If so, then in fact, .

Rings and algebras over a field

Source:

  • Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.

  • The normalizer of in a Lie ring contains the centralizer of .

  • CR(CR(S)) contains but is not necessarily equal. The double centralizer theorem deals with situations where equality occurs.

  • If is an additive subgroup of a Lie ring A, then NA(S) is the largest Lie subring of A in which is a Lie ideal.

  • If is a Lie subring of a Lie ring A, then .

See also

Notes

References