Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) 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 are 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. In mathematics, especially group theory, the centralizer (also called commutant) 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 are 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 (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. The centralizer of a subset S of group (or semigroup) G is defined to be 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 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, but if g is in the normalizer, then gs = tg for some t in S, with t potentially 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 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. 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.