A centralizer is part of an algebraic structure.
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 .