Group theory MOC

Centralizer in a group

The centralizer of an element is a Subgroup of containing all elements that commute with ,¹ 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 .

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Footnotes

1. 2017, Contemporary Abstract Algebra, p. 68 ↩