Centralizer in a group

The centralizer of an element is a Subgroup of containing all elements that commute with ,¹ #m/def/group i.e.

More generally, the centralizer of any set contains elements which commute with the whole of , i.e.

Proof of subgroups

Let . Clearly , Given any , clearly , hence is closed under the binary operation. Similarly, may be both pre- and postmultiplied by to obtain , so is closed under the inverse operation. Hence is a subgroup of by Two step subgroup test.

Since the intersection of subgroups is a subgroup, must also be a subgroup.

A related notion is the Centre of a group , which includes only those elements that commute with all group elements.

Additional properties

Clearly the centraliser itself need not be abelian, since the centraliser of any will be the entire group. For example, in the non-abelian group , the centraliser .

#state/tidy | #lang/en | #SemBr

2017, Contemporary Abstract Algebra, p. 68 ↩

Centralizer in a group

Additional properties

Footnotes