Let G a group acting on itself by conjugationMathworldPlanetmath. Let X be a subset of G. The stabilizerMathworldPlanetmath of X is called the centralizerMathworldPlanetmathPlanetmathPlanetmath of X and it’s the set

CG(X)={gG:gxg-1=xfor all xX}

For any group G, CG(G)=Z(G), the center of G. Thus, any subgroupMathworldPlanetmathPlanetmath of CG(G) is an abelianMathworldPlanetmath subgroup of G. However, the converse is generally not true. For example, take any non-abelian groupMathworldPlanetmath and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.

Title centralizer
Canonical name Centralizer1
Date of creation 2013-03-22 14:01:20
Last modified on 2013-03-22 14:01:20
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Definition
Classification msc 58E40
Related topic CentralizersInAlgebra