Definitions
- Groups and semigroups
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, however if g is in the normalizer, gs = tg for some t in S, potentially different from s. 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.
- Rings, algebras, Lie rings and Lie algebras
If R is a ring or an algebra, and S is a subset of the ring, then the centralizer of S is exactly as defined for groups, with R in the place of G.
If is a Lie algebra (or Lie ring) with Lie product, then the centralizer of a subset S of 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 = xy − yx. Of course then xy = yx if and only if = 0. 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.
The normalizer of a subset S of a Lie algebra (or Lie ring) is given by
While this is the standard usage of the term "normalizer" in Lie algebra, it should be noted that this construction is actually the idealizer of the set S in . If S is an additive subgroup of, then is the largest Lie subring (or Lie subalgebra, as the case may be) in which S is a Lie ideal.
Read more about this topic: Centralizer And Normalizer
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