The centralizer of an element
of a group
(written as
or
) is the set of elements
satisfying
More generally, let
be any subset of(not necessarily a subgroup). Then the centralizer of inis defined as
If
then
is a subgroup of
We prove the subgroup axioms one by one.
S1:
then
are in
then
so
S2:
so
S3:
then
implies
A related concept is the normalizer ofin
written as
or
The normalizer is defined as
Again,
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 subgroup
of a groupis called a self-normalizing subgroup ofif
Ifis abelian then the centralizer or normalizer of any subset ofisitself so
for any
Ifand
are any elements ofthenis in
if and only ifis in
which happens if and only ifandcommute. If
then