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:

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It is a normal subgroup (in fact, a characteristic subgroup).
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It consists of those conjugacy classes containing just one element.
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The center of an abelian group is the entire group.
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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 $\operatorname{Inn}(G)$ .