Centralizer and normalizer

In mathematics, especially group theory, the centralizer of a subset S in a group G is the set of elements of G that commute with every element of S, or equivalently, the set of elements such that conjugation by leaves each element of S fixed. The normalizer of S in G is the set of elements of G that satisfy the weaker condition of leaving the set 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. 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 ' of group G is defined as
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 is a singleton set, we write C_G instead of C_G. 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 element ''g in G'', Z.
The normalizer of S in the group G is defined as
where again only the first definition applies to semigroups. If the set is a subgroup of, then the normalizer is the largest subgroup where is a normal subgroup of. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of ' and s is in ', then it must be that, but if g is in the normalizer, then for some t in ', with t possibly different from s. That is, elements of the centralizer of ' must commute pointwise with ', 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 and both are subgroups of.

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

If R is a ring or an algebra over a field, and ' is a subset of R, then the centralizer of ' 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 ' 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 L_R, then clearly the ring centralizer of ' in R is equal to the Lie ring centralizer of ' in L_R.
The Lie bracket can also be viewed as an operation of the set ' on itself, because '. The Lie bracket makes ' a group and its centralizer would then be all elements However, since the Lie bracket is alternating, this condition is equivalent to Thus, the centralizer is defined in the same way for Lie algebras as for groups.
The normalizer of a subset ' 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 ' in. If ' is an additive subgroup of, then is the largest Lie subring in which ' is a Lie ideal.

Example

Consider the group
Take a subset of the group :
Note that is the identity permutation in and retains the order of each element and is the permutation that fixes the first element and swaps the second and third element.
The normalizer of with respect to the group are all elements of that yield the set when the element conjugates.
Working out the example for each element of :
Therefore, the normalizer of in is since both these group elements preserve the set under conjugation.
The centralizer of the group is the set of elements that leave each element of unchanged by conjugation; that is, the set of elements that commutes with every element in.
It's clear in this example that the only such element in S₃ is itself.

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 ' are both subgroups of G.
Clearly,. In fact, C_G is always a normal subgroup of N_G, being the kernel of the homomorphism and the group N_G/C_G 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, and especially if the torus is maximal need not contain '. Containment occurs exactly when ' is abelian.
If H is a subgroup of G, then N_G 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.
If ' is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains ' is the subgroup C_G.
A subgroup H of a group G is called a ' of G'' if.
The center of G is exactly C_G and G is an abelian group if and only if.
For singleton sets,.
By symmetry, if ' and T are two subsets of G, if and only if.
For a subgroup H of group G, the N/C theorem states that the factor group N_G/C_G is isomorphic to a subgroup of Aut, the group of automorphisms of H. Since and, 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 homomorphism by, then we can describe N_G and C_G in terms of the group action of Inn on G: the stabilizer of ' in Inn is T, and the subgroup of Inn fixing ' pointwise is T.
A subgroup H of a group G is said to be C-closed or self-bicommutant''' if for some subset. If so, then in fact,.
Rings and algebras over a field