Centralizer and normalizer
Special types of subgroups encountered in group theory / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Centralizer and normalizer?
Summarize this article for a 10 year old
In mathematics, especially group theory, the centralizer (also called commutant[1][2]) of a subset S in a group G is the set of elements of G that commute with every element of S, or equivalently, 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 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.