Centralizer
From AoPSWiki
A centralizer is part of an algebraic structure.
Specifically, let 
be a magma, and let 
be a subset of . The centralizer 
of is the set of elements of which commute with every element of .
If 
are subsets of a magma , then 
. The bicentralizer 
of is the centralizer of . Evidently, 
. The centralizer of the bicentralizer, 
, is equal to , for 
, but , so 
.
If the magma is associative, then the centralizer of is also the centralizer of the subset of genererated by , and the centralizer of is furthermore an associative sub-magma of . If is a group, then the centralizer of is a subgroup, though not necessarily normal. The centralizer of is also called the center of .
Centralizers in Groups
If 
is a group, then an element 
of is said to centralize 
if it commutes with every element of ; that is, if 
for all 
. A subset 
of is said to centralize if all its elements centralize . The centralizer of , denoted 
, or 
when there is no risk of confusion, is the set of elements that centralize . It is evidently a subgroup of .
