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| | Hi, I'm Colby Pennell but I never really liked that name. Credit authorising is where my primary income was inspired by. Michigan is where she and her husband live. What me and my family love would be do magic and I am going to never stop doing understand it.<br><br>My web site; [http://wiki.dayz-sa.cz/What_Is_Search_Engine_Optimization seo company in philadelphia] |
| {{Redirect|Centralizer|centralizers of Banach spaces|Multipliers and centralizers (Banach spaces)}}
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| In mathematics, especially [[group theory]], the '''centralizer''' of a [[subset]] ''S'' of a [[group (mathematics)|group]] ''G'' is the set of elements of ''G'' that [[commutativity|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 [[subgroup]]s of ''G'', and can provide insight into the structure of ''G''.
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| The definitions also apply to [[monoid]]s and [[semigroup]]s.
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| In [[ring theory]], the '''centralizer of a subset of a [[ring (mathematics)|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]].
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| The [[idealizer]] in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.
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| ==Definitions==
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| ;Groups and semigroups
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| The '''centralizer''' of a subset ''S'' of group (or semigroup) ''G'' is defined to be<ref>Jacobson (2009), p. 41</ref>
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| :<math>\mathrm{C}_G(S)=\{g\in G\mid sg=gs \text{ for all } s\in S\}</math>
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| 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 ''C''<sub>''G''</sub>({''a''}) can be abbreviated to ''C''<sub>''G''</sub>(''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'').
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| The '''normalizer''' of ''S'' in the group (or semigroup) ''G'' is defined to be
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| :<math>\mathrm{N}_G(S)=\{ g \in G \mid gS=Sg \}</math>
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| 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 [[conjugate closure|normal closure]].
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| ;Rings, algebras, Lie rings and Lie algebras
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| 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''.
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| If <math>\mathfrak{L}</math> is a [[Lie algebra]] (or [[Lie ring]]) with Lie product [''x'',''y''], then the centralizer of a subset ''S'' of <math>\mathfrak{L}</math> is defined to be{{sfn|Jacobson|1979|loc=p.28}}
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| :<math>\mathrm{C}_{\mathfrak{L}}(S)=\{ x \in \mathfrak{L} \mid [x,s]=0 \text{ for all } s\in S \}</math>
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| 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 [[commutator#(ring theory)|bracket product]] [''x'',''y''] = ''xy'' − ''yx''. Of course then ''xy'' = ''yx'' if and only if [''x'',''y''] = 0. If we denote the set ''R'' with the bracket product as ''L''<sub>''R''</sub>, then clearly the ''ring centralizer'' of ''S'' in ''R'' is equal to the ''Lie ring centralizer'' of ''S'' in ''L''<sub>''R''</sub>.
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| The normalizer of a subset ''S'' of a Lie algebra (or Lie ring) <math>\mathfrak{L}</math> is given by{{sfn|Jacobson|1979|loc=p.28}}
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| :<math>\mathrm{N}_{\mathfrak{L}}(S)=\{ x \in \mathfrak{L} \mid [x,s]\in S \text{ for all } s\in S \}</math> | |
| 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 <math>\mathfrak{L}</math>. If ''S'' is an additive subgroup of <math>\mathfrak{L}</math>, then <math>\mathrm{N}_{\mathfrak{L}}(S)</math> is the largest Lie subring (or Lie subalgebra, as the case may be) in which ''S'' is a Lie [[ideal (ring theory)|ideal]].{{sfn|Jacobson|1979|loc=p.57}}
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| ==Properties==
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| ;Groups{{sfn|Isaacs|2009|loc=Chapters 1−3}}
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| * The centralizer and normalizer of ''S'' are both subgroups of ''G''.
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| * Clearly, '''C'''<sub>''G''</sub>(S)⊆'''N'''<sub>''G''</sub>(S). In fact, '''C'''<sub>''G''</sub>(''S'') is always a [[normal subgroup]] of '''N'''<sub>''G''</sub>(''S'').
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| * '''C'''<sub>''G''</sub>('''C'''<sub>''G''</sub>(S)) contains ''S'', but '''C'''<sub>''G''</sub>(S) need not contain ''S''. Containment will occur if ''st''=''ts'' for every ''s'' and ''t'' in ''S''. Naturally then if ''H'' is an abelian subgroup of ''G'', '''C'''<sub>''G''</sub>(H) contains ''H''.
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| * If ''S'' is a subsemigroup of ''G'', then '''N'''<sub>''G''</sub>(S) contains ''S''.
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| * If ''H'' is a subgroup of ''G'', then the largest subgroup in which ''H'' is normal is the subgroup '''N'''<sub>''G''</sub>(H).
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| * A subgroup ''H'' of a group ''G'' is called a '''self-normalizing subgroup''' of ''G'' if '''N'''<sub>''G''</sub>(''H'') = ''H''.
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| * The center of ''G'' is exactly '''C'''<sub>''G''</sub>(G) and ''G'' is an [[abelian group]] if and only if '''C'''<sub>''G''</sub>(G)=Z(''G'') = ''G''.
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| * For singleton sets, '''C'''<sub>''G''</sub>(''a'')='''N'''<sub>''G''</sub>(''a'').
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| * By symmetry, if ''S'' and ''T'' are two subsets of ''G'', ''T''⊆'''C'''<sub>''G''</sub>(''S'') if and only if ''S''⊆'''C'''<sub>''G''</sub>(''T'').
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| * For a subgroup ''H'' of group ''G'', the '''N/C theorem''' states that the [[factor group]] '''N'''<sub>''G''</sub>(''H'')/'''C'''<sub>''G''</sub>(''H'') is [[group isomorphism|isomorphic]] to a subgroup of Aut(''H''), the [[automorphism group]] of ''H''. Since '''N'''<sub>''G''</sub>(''G'') = ''G'' and '''C'''<sub>''G''</sub>(''G'') = Z(''G''), the N/C theorem also implies that ''G''/Z(''G'') is isomorphic to Inn(''G''), the subgroup of Aut(''G'') consisting of all [[inner automorphism]]s of ''G''.
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| * If we define a [[group homomorphism]] ''T'' : ''G'' → Inn(''G'') by ''T''(''x'')(''g'') = ''T''<sub>''x''</sub>(''g'') = ''xgx''<sup> −1</sup>, then we can describe '''N'''<sub>''G''</sub>(''S'') and '''C'''<sub>''G''</sub>(''S'') in terms of the [[group action]] of Inn(''G'') on ''G'': the stabilizer of ''S'' in Inn(''G'') is ''T''('''N'''<sub>''G''</sub>(''S'')), and the subgroup of Inn(''G'') fixing ''S'' is ''T''('''C'''<sub>''G''</sub>(''S'')).
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| ;Rings and algebras{{sfn|Jacobson|1979|loc=p.28}}
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| * Centralizers in rings and algebras are subrings and subalgebras, respectively, and centralizers in Lie rings and Lie algebras are Lie subrings and Lie subalgebras, respectively.
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| * The normalizer of ''S'' in a Lie ring contains the centralizer of ''S''.
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| * '''C'''<sub>''R''</sub>('''C'''<sub>''R''</sub>(''S'')) contains ''S'' but is not necessarily equal. The [[double centralizer theorem]] deals with situations where equality occurs.
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| * If ''S'' is an additive subgroup of a Lie ring ''A'', then '''N'''<sub>''A''</sub>(''S'') is the largest Lie subring of ''A'' in which ''S'' is a Lie ideal.
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| * If ''S'' is a Lie subring of a Lie ring ''A'', then ''S''⊆'''N'''<sub>''A''</sub>(''S'').
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| ==See also==
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| * [[Commutant]]
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| * [[Stabilizer subgroup]]
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| * [[Multipliers and centralizers (Banach spaces)]]
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| * [[Double centralizer theorem]]
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| * [[Idealizer]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{citation |last=Isaacs |first=I. Martin |title=Algebra: a graduate course |series=Graduate Studies in Mathematics |volume=100 |edition=reprint of the 1994 original |publisher=American Mathematical Society |place=Providence, RI |year=2009 |pages=xii+516 |isbn=978-0-8218-4799-2 |mr=2472787}}
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| *{{Citation |last=Jacobson |first=Nathan |author-link=Nathan Jacobson |year=2009 |title=Basic algebra |edition=2 |volume=1 |series= |publisher=Dover |isbn=978-0-486-47189-1}}
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| *{{citation|last=Jacobson |first=Nathan |title=Lie algebras |edition=republication of the 1962 original |publisher=Dover Publications Inc. |place=New York |year=1979 |pages=ix+331 |isbn=0-486-63832-4 |mr=559927}}
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| {{DEFAULTSORT:Centralizer And Normalizer}}
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| [[Category:Abstract algebra]]
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| [[Category:Group theory]]
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| [[Category:Ring theory]]
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| [[Category:Lie algebras]]
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| [[ru:Центр группы]]
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Hi, I'm Colby Pennell but I never really liked that name. Credit authorising is where my primary income was inspired by. Michigan is where she and her husband live. What me and my family love would be do magic and I am going to never stop doing understand it.
My web site; seo company in philadelphia