Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant  ) 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 $$\mathrm{N}_G(S)$$ of G that satisfy the weaker condition of leaving the set $$S \subseteq G$$ 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 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.

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


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

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 = {a} 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


 * $$\mathrm{N}_G(S) = \left\{ g \in G \mid gS = Sg \right\} = \left\{g \in G \mid gSg^{-1} = S\right\},$$

where again only the first definition applies to semigroups. If the set $$S$$ is a subgroup of $$G$$, then the normalizer $$N_G(S)$$ is the largest subgroup $$G' \subseteq G$$ 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 gs = sg, but if g is in the normalizer, then gs = tg 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 $$C_G(S) \subseteq N_G(S)$$ 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 $$\mathfrak{L}$$ is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset S of $$\mathfrak{L}$$ is defined to be


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

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 [x, y] = xy − yx. Of course then xy = yx if and only if [x, y] = 0. 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) $$\mathfrak{L}$$ is given by
 * $$\mathrm{N}_\mathfrak{L}(S) = \{ x \in \mathfrak{L} \mid [x, s] \in S \text{ for all } s \in S \}.$$

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

Example
Consider the group
 * $$G = S_3 = \{[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\}$$ (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].

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

Groups
Source:
 * The centralizer and normalizer of S are both subgroups of G.
 * Clearly, CG(S) ⊆ NG(S). In fact, CG(S) is always a normal subgroup of NG(S), being the kernel of the homomorphism NG(S) → Bij(S) 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 W(G,T) = NG(T)/CG(T), and especially if the torus is maximal (i.e. CG(T) = T) it is a central tool in the theory of Lie groups.
 * CG(CG(S)) contains S, but CG(S) need not contain S. Containment occurs exactly when S 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 S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup CG(S).
 * A subgroup H of a group G is called a  of G if NG(H) = H.
 * The center of G is exactly CG(G) and G is an abelian group if and only if CG(G) = Z(G) = G.
 * For singleton sets, CG(a) = NG(a).
 * By symmetry, if S and T are two subsets of G, T ⊆ CG(S) if and only if S ⊆ CG(T).
 * 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 NG(G) = G and CG(G) = Z(G), 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 T : G → Inn(G) by T(x)(g) = Tx(g) = xgx−1, then we can describe NG(S) and CG(S) in terms of the group action of Inn(G) on G: the stabilizer of S in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing S pointwise is T(CG(S)).
 * A subgroup H of a group G is said to be C-closed or self-bicommutant if H = CG(S) for some subset S ⊆ G. If so, then in fact, H = CG(CG(H)).

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 S in a Lie ring contains the centralizer of S.
 * CR(CR(S)) contains S but is not necessarily equal. The double centralizer theorem deals with situations where equality occurs.
 * If S is an additive subgroup of a Lie ring A, then NA(S) is the largest Lie subring of A in which S is a Lie ideal.
 * If S is a Lie subring of a Lie ring A, then S ⊆ NA(S).