center of a group

The center of a group $G$ is the subgroup consisting of those elements that commute with every other element. Formally, $$\operatorname{Z}(G) = \{x \in G \mid xg = gx\hbox{ for all }g \in G\}.$$

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 $p$ , every non-trivial finite $p$ -group has a non-trivial center. (Proof of a stronger version of this theorem.)

A subgroup of the center of a group $G$ is called a central subgroup of $G$ . All central subgroups of $G$ are normal in $G$ .

For any group $G$ , the quotient $G/\operatorname{Z}(G)$ is called the central quotient of $G$ , and is isomorphic to the inner automorphism group $\Inn(G)$ .