# 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 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 (http://planetmath.org/PGroup4) has a non-trivial center. (Proof of a stronger version of this theorem. (http://planetmath.org/ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection))

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 (http://planetmath.org/QuotientGroup) $G/\operatorname{Z}(G)$ is called the central quotient of $G$, and is isomorphic to the inner automorphism group $\operatorname{Inn}(G)$.

Title center of a group CenterOfAGroup 2013-03-22 12:23:38 2013-03-22 12:23:38 yark (2760) yark (2760) 20 yark (2760) Definition msc 20A05 center centre CenterOfARing Centralizer central quotient