The center of a group is the subgroup consisting of those elements that commute with every other element. Formally,

It can be shown that the center has the following properties:

• It is a normal subgroup (in fact, a characteristic subgroup).
• It consists of those conjugacy classes containing just one element.
• The center of an abelian group is the entire group.
• For every prime , every non-trivial finite -group has a non-trivial center. (Proof of a stronger version of this theorem.)

A subgroup of the center of a group is called a central subgroup of .

For any group , the quotient is called the central quotient of , and is isomorphic to the inner automorphism group .