In mathematics, especially group theory, the centralizer of a subsetS of a groupG is the set of elements of G that commute with each element ofS, and the normalizer of S is the set of elements that satisfy a weaker condition. 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 operation of the ring. The centralizer of a subset of a ringR 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
Group and semigroup
The centralizer of a subset S of group G is defined as 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 instead of CG. Another less common notation for the centralizer is Z, 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, and the centralizer of an elementg in G, Z. The normalizer of S in the group G is defined as 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, 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.
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 is a Lie algebra with Lie product , then the centralizer of a subset S of 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 S in R is equal to the Lie ring centralizer of S in LR. The normalizer of a subset S of a Lie algebra 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 S in. If S is an additive subgroup of, then is the largest Lie subring in which S is a Lie ideal.
Properties
Semigroups
Let denote the centralizer of in the semigroup, i.e. Then forms a subsemigroup and, i.e. a commutant is its own bicommutant.
Groups
Source:
The centralizer and normalizer of S are both subgroups of G.
Clearly, CG ⊆ NG. In fact, CG is always a normal subgroup of NG.
CG contains S, but CG need not contain S. Containment occurs exactly when S is abelian.
If H is a subgroup of G, then NG contains H.
If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup NG.
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.
A subgroup H of a group G is called a of G if NG = H.
The center of G is exactly CG and G is an abelian group if and only if CG = Z = G.
By symmetry, if S and T are two subsets of G, T ⊆ CG if and only if S ⊆ CG.
For a subgroup H of group G, the N/C theorem states that the factor groupNG/CG is isomorphic to a subgroup of Aut, the group of automorphisms of H. Since NG = G and CG = Z, the N/C theorem also implies that G/Z is isomorphic to Inn, the subgroup of Aut consisting of all inner automorphisms of G.
If we define a group homomorphismT : G → Inn by T = Tx = xgx−1, then we can describe NG and CG in terms of the group action of Inn on G: the stabilizer of S in Inn is T, and the subgroup of Inn fixing S pointwise is T.
A subgroup H of a group G is said to be C-closed or self-bicommutant if H = CG for some subset S ⊆ G. If so, then in fact, H = CG.
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 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 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.
