## The Centralizer of a Group Element

The centralizer of an elementof a group(written asor) is the set of elementssatisfying

More generally, letbe any subset of(not necessarily a subgroup). Then the centralizer of inis defined asIfthen

is a subgroup ofWe prove the subgroup axioms one by one.

S1:thenare inthenso

S2:so

S3:thenimplies

A related concept is the normalizer ofinwritten asorThe normalizer is defined asAgain,can easily be seen to be a subgroup of The normalizer gets its name from the fact that ifis a subgroup ofthenis the largest subgroup ofwithas a normal subgroup.

A subgroupof a groupis called a self-normalizing subgroup ofif

Ifis abelian then the centralizer or normalizer of any subset ofisitself sofor any

Ifandare any elements ofthenis inif and only ifis inwhich happens if and only ifandcommute. Ifthen