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:

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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 nontrivial finite $p$group (http://planetmath.org/PGroup4) has a nontrivial 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 $\mathrm{Inn}(G)$.
Title  center of a group 

Canonical name  CenterOfAGroup 
Date of creation  20130322 12:23:38 
Last modified on  20130322 12:23:38 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  20 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20A05 
Synonym  center 
Synonym  centre 
Related topic  CenterOfARing 
Related topic  Centralizer^{} 
Defines  central quotient 