center of a group

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

$$\mathrm{Z}(G)=\{x\in G\mid xg=gx\text{ for all }g\in G\}.$$

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