Editing Centralizer and normalizer (section)

{{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.