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| {{Redirect|Normalizer|the process of increasing audio amplitude|Audio normalization}}
| | Are you looking for some info that you will want to compare afterwards? Google Squared can do this for you automatically! For instance, try it by searching it for "roller coaster" and yes, Google Squared will show you a chart of top 20 tallest roller coasters, amazing isn't it! For more, check out charts of hurricanes, damage estimates of recent hurricanes etc.<br><br>Next Google looks at what your google history reveals and attempts to work on assumptions that would be tampa seo relevant to you according to their calculations. They are doing the same thing for me.<br><br>You can choose the default value of Everything - this will do exactly as you expect and scrub all your browsing history since you've installed Firefox.<br><br>Recipes can be found easily by doing a google search, but a simple recipe includes flour, eggs and salt. Working with these ingredients using a fork until you can no longer stir the mixture is the first step. Then you would need to work with the flour by hand before feeding it into the machine. This is just a simply recipe and I am sure you have your own and most pasta machines come with recipes as well. The ladies at my church have created a recipe book and one of our Italian ladies has a great pasta recipe in the book.<br><br>Hand out 6 such deep questions. Then find solutions to the problems visitors respond with, using Google and [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=microprocessor+discussion&Submit=Go microprocessor discussion] forums. For the opinions, map out a simple tally on the various answers/brands you get in the survey. Then mark the top 3 and do some google research on those. Gather up your information. You are getting what the person needs to know or what he/she finds the best. You are extracting information. That's why this tactic is called the "EXTRACTOR".<br><br>Stop stealing photos. Yes, it is easy to do a Google [http://wiki.dayz-sa.cz/What_Is_Search_Engine_Optimization seo company in philadelphia] image search and rip pics from someone else. Unfortunately, order tampa seo that's kind of illegal. However, hope is not lost! Solicit photos from your readers. I'm sure there are Lego fans out there that would love to share their creations with you.<br><br>Cheddar's is an awesome place to take your family or even your date on the weekends. If you are planning to make a visit to Cheddar's soon, I recommend any number of these entrees. Regardless of what you order, you are going to be satisfied! |
| {{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>
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| 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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Are you looking for some info that you will want to compare afterwards? Google Squared can do this for you automatically! For instance, try it by searching it for "roller coaster" and yes, Google Squared will show you a chart of top 20 tallest roller coasters, amazing isn't it! For more, check out charts of hurricanes, damage estimates of recent hurricanes etc.
Next Google looks at what your google history reveals and attempts to work on assumptions that would be tampa seo relevant to you according to their calculations. They are doing the same thing for me.
You can choose the default value of Everything - this will do exactly as you expect and scrub all your browsing history since you've installed Firefox.
Recipes can be found easily by doing a google search, but a simple recipe includes flour, eggs and salt. Working with these ingredients using a fork until you can no longer stir the mixture is the first step. Then you would need to work with the flour by hand before feeding it into the machine. This is just a simply recipe and I am sure you have your own and most pasta machines come with recipes as well. The ladies at my church have created a recipe book and one of our Italian ladies has a great pasta recipe in the book.
Hand out 6 such deep questions. Then find solutions to the problems visitors respond with, using Google and microprocessor discussion forums. For the opinions, map out a simple tally on the various answers/brands you get in the survey. Then mark the top 3 and do some google research on those. Gather up your information. You are getting what the person needs to know or what he/she finds the best. You are extracting information. That's why this tactic is called the "EXTRACTOR".
Stop stealing photos. Yes, it is easy to do a Google seo company in philadelphia image search and rip pics from someone else. Unfortunately, order tampa seo that's kind of illegal. However, hope is not lost! Solicit photos from your readers. I'm sure there are Lego fans out there that would love to share their creations with you.
Cheddar's is an awesome place to take your family or even your date on the weekends. If you are planning to make a visit to Cheddar's soon, I recommend any number of these entrees. Regardless of what you order, you are going to be satisfied!