Centralizer and normalizer
In mathematics, especially group theory, the
centralizer 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 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.