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In mathematics, especially group theory, the centralizer (also called commutant[1][2]) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S is the set of elements of G that commute with S "as a whole". The centralizer and normalizer of S are subgroups of G, and can provide insight into the structure of G.

The definitions also apply to monoids and 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 Lie algebra.

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

Definitions

Groups and semigroups

The centralizer of a subset S of group (or semigroup) G is defined to be[3]

\( \mathrm{C}_G(S)=\{g\in G\mid sg=gs \text{ for all } s\in S\} \)

Sometimes if there is no ambiguity about the group in question, the G is suppressed from the notation entirely. When S={a} is a singleton set, then CG({a}) can be abbreviated to CG(a). Another less common notation for the centralizer is Z(a), which parallels the notation for the center of a group. 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, given by Z(g).

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

\( \mathrm{N}_G(S)=\{ g \in G \mid gS=Sg \} \)

The definitions 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, however if g is in the normalizer, gs = tg for some t in S, potentially different from s. The same conventions mentioned previously about suppressing G and suppressing braces from singleton sets also apply to the normalizer notation. The normalizer should not be confused with the normal closure.

Rings, algebras, Lie rings and Lie algebras

If R is a ring or an algebra, and S is a subset of the ring, 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[4]

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

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[4]

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

While this is the standard usage of the term "normalizer" in Lie algebra, it should be noted that 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.[5]

Properties
Semigroups

Let S′ be the centralizer, i.e. \( S'=\{x\in A: sx=xs\ \mbox{for}\ \mbox{every}\ s\in S\} \). Then:

S′ forms a subsemigroup.
S' = S''' = S''''' - A commutant is its own bicommutant.

Groups

Source: [6]

The centralizer and normalizer of S are both subgroups of G.
Clearly, C_G(S)⊆N_G(S). In fact, C_G(S) is always a normal subgroup of N_G(S).
C_G(C_G(S)) contains S, but C_G(S) need not contain S. Containment will occur if st=ts for every s and t in S. Naturally then if H is an abelian subgroup of G, C_G(H) contains H.
If S is a subsemigroup of G, then N_G(S) contains S.
If H is a subgroup of G, then the largest subgroup in which H is normal is the subgroup N_G(H).
A subgroup H of a group G is called a self-normalizing subgroup of G if N_G(H) = H.
The center of G is exactly C_G(G) and G is an abelian group if and only if C_G(G)=Z(G) = G.
For singleton sets, C_G(a)=N_G(a).
By symmetry, if S and T are two subsets of G, T⊆C_G(S) if and only if S⊆C_G(T).
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 automorphism group of H. Since N_G(G) = G and C_G(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) = T_x(g) = xgx⁻¹, 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 S in Inn(G) is T(N_G(S)), and the subgroup of Inn(G) fixing S is T(C_G(S)).
A subgroup H of a group G is said to be C-closed if H = C_G(S) for some subset S⊆G. If so, then in fact, H = C_G(C_G(H)).

Rings and algebras

Source: [4]

Centralizers in rings and algebras are subrings and subalgebras, respectively, and centralizers in Lie rings and Lie algebras are Lie subrings and Lie subalgebras, respectively.
The normalizer of S in a Lie ring contains the centralizer of S.
C_R(C_R(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 N_A(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⊆N_A(S).

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Centralizer and normalizer