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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Centralizer and normalizer ： ウィキペディア英語版 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'' 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 (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. ==Definitions== ;Groups and semigroups The centralizer of a subset ''S'' of group (or semigroup) ''G'' is defined to be〔Jacobson (2009), p. 41〕 :$\backslash mathrm\_G(S)=\backslash$ Sometimes if there is no ambiguity about the group in question, the ''G'' is suppressed from the notation entirely. When ''S''= is a singleton set, then C''G''() can be abbreviated to C''G''(''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 :$\backslash mathrm\_G(S)=\backslash$ 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 , however if ''g'' is in the normalizer, 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 $\backslash mathfrak$ is a Lie algebra (or Lie ring) with Lie product (), then the centralizer of a subset ''S'' of $\backslash mathfrak$ is defined to be :$\backslash mathrm\_\; \backslash mid\; ()=0\; \backslash text\; s\backslash in\; S\; \backslash \}$ 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 . Of course then if and only if . If we denote the set ''R'' with the bracket product as L''R'', then clearly the ''ring centralizer'' of ''S'' in ''R'' is equal to the ''Lie ring centralizer'' of ''S'' in L''R''. The normalizer of a subset ''S'' of a Lie algebra (or Lie ring) $\backslash mathfrak$ is given by :$\backslash mathrm\_\; \backslash mid\; ()\backslash in\; S\; \backslash text\; s\backslash in\; S\; \backslash \}$ 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 $\backslash mathfrak$. If ''S'' is an additive subgroup of $\backslash mathfrak$, then $\backslash mathrm\_\{\backslash mathfrak\{L\}\}(S)$ is the largest Lie subring (or Lie subalgebra, as the case may be) in which ''S'' is a Lie ideal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Centralizer and normalizer」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.