Editing Centralizer and normalizer (section)

==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'')}}.