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.
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