Editing Centralizer and normalizer

{{Short description|Special types of subgroups encountered in group theory}}
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In [[mathematics]], especially [[group theory]], the '''centralizer'''  (also called '''commutant'''<ref name="O'MearaClark2011">{{cite book|author1=Kevin O'Meara|author2=John Clark|author3=Charles Vinsonhaler|title=Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form|url=https://books.google.com/books?id=HLiWsnzJe6MC&pg=PA65|year=2011|publisher= [[Oxford University Press]]|isbn=978-0-19-979373-0|page=65}}</ref><ref name="HofmannMorris2007">{{cite book|author1=Karl Heinrich Hofmann|author2=Sidney A. Morris|title=The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups|url=https://books.google.com/books?id=fJyqSkEexNgC&pg=PA30|year=2007|publisher= [[European Mathematical Society]]|isbn=978-3-03719-032-6|page=30}}</ref>) of a [[subset]] ''S'' in a [[group (mathematics)|group]] ''G'' is the set <math>\operatorname{C}_G(S)</math> of elements of ''G'' that [[commutativity|commute]] with every element of ''S'', or equivalently, the set of elements <math>g\in G</math> such that [[Conjugation (group theory)|conjugation]] by <math>g</math> leaves each element of ''S'' fixed. The '''normalizer''' of ''S'' in ''G'' is the [[Set (mathematics)|set]] of elements <math>\mathrm{N}_G(S)</math> of ''G'' that satisfy the weaker condition of leaving the set <math>S \subseteq G</math> fixed under conjugation. The centralizer and normalizer of ''S'' are [[subgroup]]s of ''G''.  Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets&nbsp;''S''.

Suitably formulated, the definitions also apply to [[semigroup]]s.

In [[ring theory]], the '''centralizer of a subset of a [[ring (mathematics)|ring]]''' is defined with respect to the multiplication of the ring (a semigroup operation). 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.

==Definitions==
===Group and semigroup===
The '''centralizer''' of a subset ''<math>S</math>'' of group (or semigroup) ''G'' is defined as<ref>Jacobson (2009), p. 41</ref>

:<math>\mathrm{C}_G(S) = \left\{g \in G \mid gs = sg \text{ for all } s \in S\right\} = \left\{g \in G \mid gsg^{-1} = s \text{ for all } s \in S\right\},</math>

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. When <math>S=\{a\}</math> is a [[singleton (mathematics)|singleton]] set, we write C<sub>''G''</sub>(''a'') instead of C<sub>''G''</sub>({''a''}). Another less common notation for the centralizer is Z(''a''), which parallels the notation for the [[Center (group theory)|center]].  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

:<math>\mathrm{N}_G(S) = \left\{ g \in G \mid gS = Sg \right\} = \left\{g \in G \mid gSg^{-1} = S\right\},</math>

where again only the first definition applies to semigroups. If the set <math>S</math> is a subgroup of <math>G</math>, then the normalizer <math>N_G(S)</math> is the largest subgroup <math>G' \subseteq G</math> where <math>S</math> is a [[normal subgroup]] of <math>G'</math>. The definitions of ''centralizer'' and ''normalizer'' are similar but not identical. If ''g'' is in the centralizer of ''<math>S</math>'' and ''s'' is in ''<math>S</math>'', then it must be that {{nowrap|1=''gs'' = ''sg''}}, but if ''g'' is in the normalizer, then {{nowrap|1=''gs'' = ''tg''}} for some ''t'' in ''<math>S</math>'', with ''t'' possibly different from ''s''. That is, elements of the centralizer of ''<math>S</math>'' must commute pointwise with ''<math>S</math>'', 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. The normalizer should not be confused with the [[conjugate closure|normal closure]].

Clearly <math>C_G(S) \subseteq N_G(S)</math> and both are subgroups of <math>G</math>.

===Ring, algebra over a field, Lie ring, and Lie algebra===
If ''R'' is a ring or an [[algebra over a field]], and ''<math>S</math>'' is a subset of ''R'', then the centralizer of ''<math>S</math>'' is exactly as defined for groups, with ''R'' in the place of ''G''.

If <math>\mathfrak{L}</math> is a [[Lie algebra]] (or [[Lie ring]]) with Lie product [''x'', ''y''], then the centralizer of a subset ''<math>S</math>'' of <math>\mathfrak{L}</math> is defined to be{{sfn|Jacobson|1979|loc=p. 28}}

:<math>\mathrm{C}_{\mathfrak{L}}(S) = \{ x \in \mathfrak{L} \mid [x, s] = 0 \text{ for all } s \in S \}.</math>
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]] {{nowrap|1=[''x'', ''y''] = ''xy'' − ''yx''}}. Of course then {{nowrap|1=''xy'' = ''yx''}} if and only if {{nowrap|1=[''x'', ''y''] = 0}}. If we denote the set ''R'' with the bracket product as L<sub>''R''</sub>, then clearly the ''ring centralizer'' of ''<math>S</math>'' in ''R'' is equal to the ''Lie ring centralizer'' of ''<math>S</math>''  in L<sub>''R''</sub>.

The normalizer of a subset ''<math>S</math>'' of a Lie algebra (or Lie ring) <math>\mathfrak{L}</math> is given by{{sfn|Jacobson|1979|loc=p. 28}}
:<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, this construction is actually the [[idealizer]] of the set ''<math>S</math>'' in <math>\mathfrak{L}</math>. If ''<math>S</math>'' 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 ''<math>S</math>'' is a Lie [[ideal (ring theory)|ideal]].{{sfn|Jacobson|1979|loc=p. 57}}

==Example==
Consider the group 
:<math>G = S_3 = \{[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\}</math> (the symmetric group of permutations of 3 elements).

Take a subset <math>H</math> of the group <math>G</math>: 

:<math>H = \{[1, 2, 3], [1, 3, 2]\}. </math>

Note that <math>[1, 2, 3]</math> is the identity permutation in <math>G</math> and retains the order of each element and <math>[1, 3, 2]</math> is the permutation that fixes the first element and swaps the second and third element.

The normalizer of <math>H</math> with respect to the group <math>G</math> are all elements of <math>G</math> that yield the set <math>H</math> (potentially permuted) when the element conjugates <math>H</math>.
Working out the example for each element of <math>G</math>:

:<math>[1, 2, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 2, 3]</math> is in the normalizer <math>N_G(H)</math>.
:<math>[1, 3, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [1, 3, 2]\} = H</math>; therefore <math>[1, 3, 2]</math> is in the normalizer <math>N_G(H)</math>.
:<math>[2, 1, 3]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[2, 1, 3]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[2, 3, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[2, 3, 1]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[3, 1, 2]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [3, 2, 1]\} \neq H</math>; therefore <math>[3, 1, 2]</math> is not in the normalizer <math>N_G(H)</math>.
:<math>[3, 2, 1]</math> when applied to <math>H</math>: <math>\{[1, 2, 3], [2, 1, 3]\} \neq H</math>; therefore <math>[3, 2, 1]</math> is not in the normalizer <math>N_G(H)</math>.

Therefore, the normalizer <math>N_G(H)</math> of <math>H</math> in <math>G</math> is <math>\{[1, 2, 3], [1, 3, 2]\}</math> since both these group elements preserve the set <math>H</math> under conjugation.

The centralizer of the group <math>G</math> is the set of elements that leave each element of <math>H</math> unchanged by conjugation; that is, the set of elements that commutes with every element in <math>H</math>. 
It's clear in this example that the only such element in S<sub>3</sub> is <math>H</math> itself ([1, 2, 3], [1, 3, 2]).

==Properties==
===Semigroups===
Let <math>S'</math> denote the centralizer of <math>S</math> in the semigroup <math>A</math>; i.e. <math>S' = \{x \in A \mid sx = xs \text{ for every } s \in S\}.</math> Then <math>S'</math> forms a [[subsemigroup]] and <math>S' = S''' = S'''''</math>; i.e. a commutant is its own [[bicommutant]].

===Groups===
Source:{{sfn|Isaacs|2009|loc=Chapters 1−3}}
* The centralizer and normalizer of ''<math>S</math>'' are both subgroups of ''G''.
* Clearly, {{nowrap|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''), being the kernel of the [[group homomorphism|homomorphism]] {{nowrap|N<sub>''G''</sub>(''S'') → Bij(''S'')}} and the group N<sub>''G''</sub>(''S'')/C<sub>''G''</sub>(''S'') [[Group action (mathematics)|acts]] by conjugation as a [[Symmetric group|group of bijections]] on ''S''. E.g. the [[Weyl group]] of a compact [[Lie group]] ''G'' with a torus ''T'' is defined as {{nowrap|1=''W''(''G'',''T'') = N<sub>''G''</sub>(''T'')/C<sub>''G''</sub>(''T'')}}, and especially if the torus is maximal (i.e. {{nowrap|1=C<sub>''G''</sub>(''T'') = ''T'')}} it is a central tool in the theory of Lie groups. 
* C<sub>''G''</sub>(C<sub>''G''</sub>(''S'')) contains ''<math>S</math>'', but C<sub>''G''</sub>(''S'') need not contain ''<math>S</math>''. Containment occurs exactly when ''<math>S</math>'' is abelian.
* If ''H'' is a subgroup of ''G'', then N<sub>''G''</sub>(''H'') contains ''H''.
* If ''H'' is a subgroup of ''G'', then the largest subgroup of ''G'' in which ''H'' is normal is the subgroup N<sub>''G''</sub>(''H'').
* If ''<math>S</math>'' is a subset of ''G'' such that all elements of ''S'' commute with each other, then the largest subgroup of ''G'' whose center contains ''<math>S</math>'' is the subgroup C<sub>''G''</sub>(''S'').
* A subgroup ''H'' of a group ''G'' is called a '''{{visible anchor|self-normalizing subgroup}}''' of ''G'' if {{nowrap|1=N<sub>''G''</sub>(''H'') = ''H''}}.
* The center of ''G'' is exactly  C<sub>''G''</sub>(G) and ''G'' is an [[abelian group]] if and only if {{nowrap|1=C<sub>''G''</sub>(G) = Z(''G'') = ''G''}}.
* For singleton sets, {{nowrap|1=C<sub>''G''</sub>(''a'') = N<sub>''G''</sub>(''a'')}}.
* By symmetry, if ''<math>S</math>'' and ''T'' are two subsets of ''G'', {{nowrap|''T'' ⊆ C<sub>''G''</sub>(''S'')}} if and only if {{nowrap|''S'' ⊆ C<sub>''G''</sub>(''T'')}}.
* 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 group of [[automorphism]]s of ''H''. Since {{nowrap|1=N<sub>''G''</sub>(''G'') = ''G''}} and {{nowrap|1=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''.
* If we define a group homomorphism {{nowrap|''T'' : ''G'' → Inn(''G'')}} by {{nowrap|1=''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 ''<math>S</math>'' in Inn(''G'') is ''T''(N<sub>''G''</sub>(''S'')), and the subgroup of Inn(''G'') fixing ''<math>S</math>'' pointwise is ''T''(C<sub>''G''</sub>(''S'')).
* A subgroup ''H'' of a group ''G'' is said to be '''C-closed''' or '''self-bicommutant''' if {{nowrap|1=''H'' = C<sub>''G''</sub>(''S'')}} for some subset {{nowrap|''S'' ⊆ ''G''}}. If so, then in fact, {{nowrap|1=''H'' = C<sub>''G''</sub>(C<sub>''G''</sub>(''H''))}}.

===Rings and algebras over a field===
Source:{{sfn|Jacobson|1979|loc=p. 28}}
* Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.
* The normalizer of ''<math>S</math>'' in a Lie ring contains the centralizer of ''<math>S</math>''.
* C<sub>''R''</sub>(C<sub>''R''</sub>(''S'')) contains ''<math>S</math>'' but is not necessarily equal. The [[double centralizer theorem]] deals with situations where equality occurs.
* If ''<math>S</math>'' is an additive subgroup of a Lie ring ''A'', then N<sub>''A''</sub>(''S'') is the largest Lie subring of ''A'' in which ''<math>S</math>'' is a Lie ideal.
* If ''<math>S</math>'' is a Lie subring of a Lie ring ''A'', then {{nowrap|''S'' ⊆ N<sub>''A''</sub>(''S'')}}.

==See also==
* [[Commutator]]
* [[Multipliers and centralizers (Banach spaces)]]
* [[Stabilizer subgroup]]

==Notes==
<references/>

==References==
*{{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 |isbn=978-0-8218-4799-2 |mr=2472787 |doi=10.1090/gsm/100|url=https://books.google.com/books?id=5tKq0kbHuc4C&q=centralizer+OR+normalizer|url-access=subscription }}
*{{Citation |last=Jacobson |first=Nathan |author-link=Nathan Jacobson |year=2009 |title=Basic Algebra |edition=2 |volume=1 |publisher= [[Dover Publications]] |isbn=978-0-486-47189-1|url=https://books.google.com/books?id=qAg_AwAAQBAJ&q=centralizer+OR+normalizer}}
*{{citation|last=Jacobson |first=Nathan |title=Lie Algebras |edition=republication of the 1962 original |publisher= [[Dover Publications]] |year=1979  |isbn=0-486-63832-4  |mr=559927|url=https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer}}

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[[Category:Ring theory]]
[[Category:Lie algebras]]