center of a group

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

Z(G)={xGxg=gx for all gG}.

It can be shown that the center has the following properties:

  • It consists of those conjugacy classesMathworldPlanetmathPlanetmath containing just one element.

  • The center of an abelian groupMathworldPlanetmath 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 quotientPlanetmathPlanetmath ( G/Z(G) is called the central quotient of G, and is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the inner automorphism group Inn(G).

Title center of a group
Canonical name CenterOfAGroup
Date of creation 2013-03-22 12:23:38
Last modified on 2013-03-22 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 CentralizerMathworldPlanetmath
Defines central quotient