Let G be a group, and let HG. The normalizerMathworldPlanetmathPlanetmath of H in G, written NG(H), is the set


A subgroupMathworldPlanetmathPlanetmath H of G is said to be self-normalizing if NG(H)=H.


NG(H) is always a subgroup of G, as it is the stabilizerMathworldPlanetmath of H under the action (g,H)gHg-1 of G on the set of all subsets of G (or on the set of all subgroups of G, if H is a subgroup).

If H is a subgroup of G, then HNG(H).

If H is a subgroup of G, then H is a normal subgroupMathworldPlanetmath of NG(H); in fact, NG(H) is the largest subgroup of G of which H is a normal subgroup. In particular, if H is a subgroup of G, then H is normal in G if and only if NG(H)=G.

Title normalizer
Canonical name Normalizer
Date of creation 2013-03-22 12:36:53
Last modified on 2013-03-22 12:36:53
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Definition
Classification msc 20A05
Synonym normaliser
Related topic CentralizerMathworldPlanetmathPlanetmath
Related topic NormalSubgroup
Related topic NormalClosure2
Defines self-normalizing