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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 ''S'' 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 ''S''&nbsp;=&nbsp;{''a''} 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 ''S'' and ''s'' is in ''S'', 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 ''S'', with ''t'' possibly different from ''s''. That is, elements of the centralizer of ''S'' must commute pointwise with ''S'', 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 ''S'' is a subset of ''R'', then the centralizer of ''S'' 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 ''S'' 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 ''S'' in ''R'' is equal to the ''Lie ring centralizer'' of ''S''  in L<sub>''R''</sub>.

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}}
:<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 ''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}}

==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 H of the group G: 

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

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

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

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

Therefore, the Normalizer(H) with respect to G is <math>\{[1, 2, 3], [1, 3, 2]\}</math> since both these group elements preserve the set H under conjugation.

The centralizer of the group G is the set of elements that leave each element of H unchanged by conjugation; that is, the set of elements that commutes with every element in H. 
It's clear in this example that the only such element in S<sub>3</sub> is H 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 ''S'' 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 homomorphism {{nowrap|N<sub>''G''</sub>(''S'') → Bij(''S'')}} and the group N<sub>''G''</sub>(''S'')/C<sub>''G''</sub>(''S'') acts by conjugation as a 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 ''S'', but C<sub>''G''</sub>(''S'') need not contain ''S''. Containment occurs exactly when ''S'' 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 ''S'' is a subset of ''G'' such that all elements of ''S'' commute with each other, then the largest subgroup of ''G'' whose center contains ''S'' 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 ''S'' 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 (mathematics)|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'' 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 ''S'' in a Lie ring contains the centralizer of ''S''.
* 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.
* 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.
* If ''S'' is a Lie subring of a Lie ring ''A'', then {{nowrap|''S'' ⊆ N<sub>''A''</sub>(''S'')}}.

==See also==
* [[Commutator]]
* [[Double centralizer theorem]]
* [[Idealizer]]
* [[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}}
*{{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:Abstract algebra]]
[[Category:Group theory]]
[[Category:Ring theory]]
[[Category:Lie algebras]]