Centralizer and normalizer explained

In mathematics, especially group theory, the centralizer (also called commutant[1] [2]) of a subset S in a group G is the set

\operatorname{C}G(S)

of elements of G that commute with every element of S, or equivalently, such that conjugation by

g

leaves each element of S fixed. The normalizer of S in G is the set of elements

NG(S)

of G that satisfy the weaker condition of leaving the set

S\subseteqG

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 subsets S.

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 semigroup (multiplication) operation of the ring. 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 S of group (or semigroup) G is defined as[3]

CG(S)=\left\{g\inG\midgs=sgforalls\inS\right\}=\left\{g\inG\midgsg-1=sforalls\inS\right\},

where only the first definition applies to semigroups.If there is no ambiguity about the group in question, the G can be suppressed from the notation. When S =  is a singleton set, we write CG(a) instead of CG. 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

NG(S)=\left\{g\inG\midgS=Sg\right\}=\left\{g\inG\midgSg-1=S\right\},

where again only the first definition applies to semigroups. If the set

S

is a subgroup of

G

, then the normalizer

NG(S)

is the largest subgroup

G'\subseteqG

where

S

is a normal subgroup of

G'

. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of S and s is in S, then it must be that, but if g is in the normalizer, then for some t in S, with t possibly different from s. That is, elements of the centralizer of S must commute pointwise with S, 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.

Clearly

CG(S)\subseteqNG(S)

and both are subgroups of

G

.

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

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

If

ak{L}

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

ak{L}

is defined to be

Cak{L

}(S) = \.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 S in R is equal to the Lie ring centralizer of S in LR.

The normalizer of a subset S of a Lie algebra (or Lie ring)

ak{L}

is given by

Nak{L}(S)=\{x\inak{L}\mid[x,s]\inSforalls\inS\}.

While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set S in

ak{L}

. If S is an additive subgroup of

ak{L}

, then

Nak{L

}(S) is the largest Lie subring (or Lie subalgebra, as the case may be) in which S is a Lie ideal.

Example

Consider the group

G=S3=\{[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]\}

(the symmetric group of permutations of 3 elements).

Take a subset H of the group G:

H=\{[1,2,3],[1,3,2]\}.

Note that [1, 2, 3] is the identity permutation in G and retains the order of each element and [1, 3, 2] is the permutation that fixes the first element and swaps the second and third element.

The normalizer of H with respect to the group G are all elements of G that yield the set H (potentially permuted) when the group operation is applied.Working out the example for each element of G:

[1,2,3]

when applied to H =>

\{[1,2,3],[1,3,2]\}=H

; therefore [1, 2, 3] is in the Normalizer(H) with respect to G.

[1,3,2]

when applied to H =>

\{[1,3,2],[1,2,3]\}=H

; therefore [1, 3, 2] is in the Normalizer(H) with respect to G.

[2,1,3]

when applied to H =>

\{[2,1,3],[3,1,2]\}!=H

; therefore [2, 1, 3] is not in the Normalizer(H) with respect to G.

[2,3,1]

when applied to H =>

\{[2,3,1],[3,2,1]\}!=H

; therefore [2, 3, 1] is not in the Normalizer(H) with respect to G.

[3,1,2]

when applied to H =>

\{[3,1,2],[2,1,3]\}!=H

; therefore [3, 1, 2] is not in the Normalizer(H) with respect to G.

[3,2,1]

when applied to H =>

\{[3,2,1],[2,3,1]\}!=H

; therefore [3, 2, 2] is not in the Normalizer(H) with respect to G.

Therefore, the Normalizer(H) with respect to G is

\{[1,2,3],[1,3,2]\}

since both these group elements preserve the set H.

A group is considered simple if the normalizer with respect to a subset is always the identity and itself. Here, it's clear that S3 is not a simple group.

The centralizer of the group G is the set of elements that leave each element of H unchanged. It's clear that the only such element in S3 is the identity element [1, 2, 3].

Properties

Semigroups

Let

S'

denote the centralizer of

S

in the semigroup

A

; i.e.

S'=\{x\inA\midsx=xsforeverys\inS\}.

Then

S'

forms a subsemigroup and

S'=S'''=S'''''

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

Groups

Source:

Rings and algebras over a field

Source:

See also

Notes

  1. Book: Kevin O'Meara. John Clark. Charles Vinsonhaler. Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form. 2011. Oxford University Press. 978-0-19-979373-0. 65.
  2. Book: Karl Heinrich Hofmann. Sidney A. Morris. The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups. 2007. European Mathematical Society. 978-3-03719-032-6. 30.
  3. Jacobson (2009), p. 41