Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset "S" in a group "G" is the set of elements formula_1 of "G" such that each member formula_2 commutes with each element of "S", or equivalently, such that conjugation by formula_3 leaves each element of "S" fixed. The normalizer of "S" in "G" is the set of elements formula_4 of "G" that satisfy the weaker condition of leaving the set formula_5 fixed under conjugation. The centralizer and normalizer of "S" are subgroups of "G". Many techniques in group theory are based on studying the centralisers and normalisers of suitable subsets "S".
Suitably formulated, 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 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 "S" of group (or semigroup) "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 "S" = {"a"} 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
where again only the first definition applies to semigroups. 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.
Clearly formula_8 and both are subgroups of formula_9. 
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 formula_10 is a Lie algebra (or Lie ring) with Lie product ["x","y"], then the centralizer of a subset "S" of formula_10 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 "S" in "R" is equal to the "Lie ring centralizer" of "S" in L"R".
The normalizer of a subset "S" of a Lie algebra (or Lie ring) formula_10 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 formula_10. If "S" is an additive subgroup of formula_10, then formula_17 is the largest Lie subring (or Lie subalgebra, as the case may be) in which "S" is a Lie ideal.
Properties.
Semigroups.
Let formula_18 denote the centralizer of formula_19 in the semigroup formula_20, i.e. formula_21 Then formula_18 forms a subsemigroup and formula_23, i.e. a commutant is its own bicommutant.
Groups.
Source:
Rings and algebras over a field.
Source: