In mathematics, especially group theory, the centralizer (also called commutant[1][2]) of a subset S in a group G is the set
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
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.
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.
The centralizer of a subset S of group (or semigroup) G is defined as[3] 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.
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 of centralizer and normalizer 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 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.
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
is defined to be[4] 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 [x, y] = xy − yx.
If we denote the set R with the bracket product as LR, then clearly the ring centralizer of S in R is equal to the Lie ring centralizer of S in LR.
is given by[4] While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set S in
[5] Consider the group Take a subset H of the group G: Note that [1, 2, 3] is the identity permutation in G and retains the order of each element and [1, 3, 2] is the permutation that fixes the first element and swaps the second and third element.
The normalizer of H with respect to the group G are all elements of G that yield the set H (potentially permuted) when the element conjugates H. Working out the example for each element of G: Therefore, the Normalizer(H) with respect to G is
since both these group elements preserve the set H under conjugation.