Centralizer and normalizer

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially <a href="/facts/Group_theory/NfowcI5H">group theory</a>, the centralizer (also called commutant) of a <a href="/facts/Subset/SJGERmbA">subset</a> S in a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G is the set 
 
 
 
 
 C
 
 G
 
 
 ⁡
 (
 S
 )
 
 
 {\displaystyle \operatorname {C} _{G}(S)}
 
 of elements of G that <a href="/facts/Commutativity/WaYxJLxd">commute</a> with every element of S, or equivalently, the set of elements 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
 such that <a href="/facts/Conjugation_(group_theory)/AFuPIFNW">conjugation</a> by 
 
 
 
 g
 
 
 {\displaystyle g}
 
 leaves each element of S fixed. The normalizer of S in G is the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of elements 
 
 
 
 
 
 N
 
 
 G
 
 
 (
 S
 )
 
 
 {\displaystyle \mathrm {N} _{G}(S)}
 
 of G that satisfy the weaker condition of leaving the set 
 
 
 
 S
 ⊆
 G
 
 
 {\displaystyle S\subseteq G}
 
 fixed under conjugation. The centralizer and normalizer of S are <a href="/facts/Subgroup/r91mBgpa">subgroups</a> of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S.
Suitably formulated, the definitions also apply to <a href="/facts/Semigroup/rgSdk2kt">semigroups</a>.
In <a href="/facts/Ring_theory/VW6svEFe">ring theory</a>, the centralizer of a subset of a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring R is a <a href="/facts/Subring/M0perybt">subring</a> of R. This article also deals with centralizers and normalizers in a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>.
The <a href="/facts/Idealizer/A6xVTD4m">idealizer</a> in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

Centralizer and normalizer open-in-new

Centralizer and normalizer