Centralizer

That is to say G a group, and S part of G . The centralizing of S in G , noted C _G ( S ) (or C ( S ) if the context is not ambiguous) is the sub-group of G elements which commutate with all the elements of S . If S is a singleton { has }, its centralizer is simply noted C ( has ), and if S is G entire his centralizer is called center G and is noted Z ( G ) (in reference to the German word Zentrum ). In symbols: C ( S ) = {X ∈ G ; ∀ S ∈ S sx = xs }.

Properties

The centralizer of S returns S central in a certain way: if there exists a sub-group H of G such as S is included in the center of H , then H is included in the centralizer of S . In other words, the centralizer of S is tallest (within the meaning of inclusion) sub-group of G whose center contains S , provided that such a sub-group exists indeed (it is always the case if S is a singleton).

In the general case, the centralizer of S is a Sous-groupe distinguished from the Normalisateur of S . One can thus consider the group quotient NR ( S )/ C ( S ), which one shows if S is a group which it is isomorphous with Int ( S ), the group of the interior automorphisms of S .

Random links: