subgroups containing the normalizers of Sylow subgroups normalize themselves

Let G be a finite groupMathworldPlanetmath, and S a Sylow subgroup. Let M be a subgroupMathworldPlanetmathPlanetmath such that NG(S)M. Then M=NG(M).


By order considerations, S is a Sylow subgroup of M. Since M is normal in NG(M), by the Frattini argument, NG(M)=NG(S)M=M. ∎

Title subgroups containing the normalizersMathworldPlanetmath of Sylow subgroups normalize themselves
Canonical name SubgroupsContainingTheNormalizersOfSylowSubgroupsNormalizeThemselves
Date of creation 2013-03-22 13:16:47
Last modified on 2013-03-22 13:16:47
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 6
Author bwebste (988)
Entry type Corollary
Classification msc 20D20