Centralizer and normalizer

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In mathematics, especially group theory, the centralizer (also called commutant<ref name="O'MearaClark2011">Template:Cite book</ref><ref name="HofmannMorris2007">Template:Cite book</ref>) of a subset S in a group G is the set <math>\operatorname{C}_G(S)</math> of elements of G that commute with every element of S, or equivalently, the set of elements <math>g\in G</math> such that conjugation by <math>g</math> leaves each element of S fixed. The normalizer of S in G is the set of elements <math>\mathrm{N}_G(S)</math> of G that satisfy the weaker condition of leaving the set <math>S \subseteq G</math> 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 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.

DefinitionsEdit

Group and semigroupEdit

The centralizer of a subset <math>S</math> of group (or semigroup) G is defined as<ref>Jacobson (2009), p. 41</ref>

<math>\mathrm{C}_G(S) = \left\{g \in G \mid gs = sg \text{ for all } s \in S\right\} = \left\{g \in G \mid gsg^{-1} = s \text{ for all } s \in S\right\},</math>

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 <math>S=\{a\}</math> is a singleton set, we write C_G(a) instead of C_G({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

<math>\mathrm{N}_G(S) = \left\{ g \in G \mid gS = Sg \right\} = \left\{g \in G \mid gSg^{-1} = S\right\},</math>

where again only the first definition applies to semigroups. If the set <math>S</math> is a subgroup of <math>G</math>, then the normalizer <math>N_G(S)</math> is the largest subgroup <math>G' \subseteq G</math> where <math>S</math> is a normal subgroup of <math>G'</math>. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of <math>S</math> and s is in <math>S</math>, then it must be that Template:Nowrap, but if g is in the normalizer, then Template:Nowrap for some t in <math>S</math>, with t possibly different from s. That is, elements of the centralizer of <math>S</math> must commute pointwise with <math>S</math>, 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 <math>C_G(S) \subseteq N_G(S)</math> and both are subgroups of <math>G</math>.

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

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

If <math>\mathfrak{L}</math> is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset <math>S</math> of <math>\mathfrak{L}</math> is defined to beTemplate:Sfn

<math>\mathrm{C}_{\mathfrak{L}}(S) = \{ x \in \mathfrak{L} \mid [x, s] = 0 \text{ for all } s \in S \}.</math>

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 Template:Nowrap. Of course then Template:Nowrap if and only if Template:Nowrap. If we denote the set R with the bracket product as L_R, then clearly the ring centralizer of <math>S</math> in R is equal to the Lie ring centralizer of <math>S</math> in L_R.

The normalizer of a subset <math>S</math> of a Lie algebra (or Lie ring) <math>\mathfrak{L}</math> is given byTemplate:Sfn

<math>\mathrm{N}_\mathfrak{L}(S) = \{ x \in \mathfrak{L} \mid [x, s] \in S \text{ for all } s \in S \}.</math>

While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set <math>S</math> in <math>\mathfrak{L}</math>. If <math>S</math> is an additive subgroup of <math>\mathfrak{L}</math>, then <math>\mathrm{N}_{\mathfrak{L}}(S)</math> is the largest Lie subring (or Lie subalgebra, as the case may be) in which <math>S</math> is a Lie ideal.Template:Sfn

ExampleEdit

Consider the group

<math>G = S_3 = \{[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\}</math> (the symmetric group of permutations of 3 elements).

Take a subset <math>H</math> of the group <math>G</math>:

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

The normalizer of <math>H</math> with respect to the group <math>G</math> are all elements of <math>G</math> that yield the set <math>H</math> (potentially permuted) when the element conjugates <math>H</math>. Working out the example for each element of <math>G</math>:

<math>[1, 2, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 2, 3]</math> is in the normalizer <math>N_G(H)</math>.

<math>[1, 3, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 3, 2]</math> is in the normalizer <math>N_G(H)</math>.

<math>[2, 1, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[2, 1, 3]</math> is not in the normalizer <math>N_G(H)</math>.

<math>[2, 3, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[2, 3, 1]</math> is not in the normalizer <math>N_G(H)</math>.

<math>[3, 1, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[3, 1, 2]</math> is not in the normalizer <math>N_G(H)</math>.

<math>[3, 2, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[3, 2, 1]</math> is not in the normalizer <math>N_G(H)</math>.

Therefore, the normalizer <math>N_G(H)</math> of <math>H</math> in <math>G</math> is <math>\{[1, 2, 3], [1, 3, 2]\}</math> since both these group elements preserve the set <math>H</math> under conjugation.

The centralizer of the group <math>G</math> is the set of elements that leave each element of <math>H</math> unchanged by conjugation; that is, the set of elements that commutes with every element in <math>H</math>. It's clear in this example that the only such element in S₃ is <math>H</math> itself ([1, 2, 3], [1, 3, 2]).

PropertiesEdit

SemigroupsEdit

Let <math>S'</math> denote the centralizer of <math>S</math> in the semigroup <math>A</math>; i.e. <math>S' = \{x \in A \mid sx = xs \text{ for every } s \in S\}.</math> Then <math>S'</math> forms a subsemigroup and <math>S' = S = S</math>; i.e. a commutant is its own bicommutant.

GroupsEdit

Source:Template:Sfn

The centralizer and normalizer of <math>S</math> are both subgroups of G.
Clearly, Template:Nowrap. In fact, C_G(S) is always a normal subgroup of N_G(S), being the kernel of the homomorphism Template:Nowrap and the group N_G(S)/C_G(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 Template:Nowrap, and especially if the torus is maximal (i.e. Template:Nowrap it is a central tool in the theory of Lie groups.
C_G(C_G(S)) contains <math>S</math>, but C_G(S) need not contain <math>S</math>. Containment occurs exactly when <math>S</math> is abelian.
If H is a subgroup of G, then N_G(H) contains H.
If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N_G(H).
If <math>S</math> is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains <math>S</math> is the subgroup C_G(S).
A subgroup H of a group G is called a Template:Visible anchor of G if Template:Nowrap.
The center of G is exactly C_G(G) and G is an abelian group if and only if Template:Nowrap.
For singleton sets, Template:Nowrap.
By symmetry, if <math>S</math> and T are two subsets of G, Template:Nowrap if and only if Template:Nowrap.
For a subgroup H of group G, the N/C theorem states that the factor group N_G(H)/C_G(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H. Since Template:Nowrap and Template:Nowrap, 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 Template:Nowrap by Template:Nowrap, then we can describe N_G(S) and C_G(S) in terms of the group action of Inn(G) on G: the stabilizer of <math>S</math> in Inn(G) is T(N_G(S)), and the subgroup of Inn(G) fixing <math>S</math> pointwise is T(C_G(S)).
A subgroup H of a group G is said to be C-closed or self-bicommutant if Template:Nowrap for some subset Template:Nowrap. If so, then in fact, Template:Nowrap.

Rings and algebras over a fieldEdit