normality of subgroups of prime index


If H is a subgroupMathworldPlanetmathPlanetmath of a finite groupMathworldPlanetmath G of index p, where p is the smallest prime dividing the order of G, then H is normal in G.


Suppose HG with |G| finite and |G:H|=p, where p is the smallest prime divisorPlanetmathPlanetmath of |G|, let G act on the set L of left cosetsMathworldPlanetmath of H in G by left , and let φ:GSp be the induced by this action. Now, if gkerφ, then gxH=xH for each xG, and in particular, gH=H, whence gH. Thus K=kerφ is a normal subgroupMathworldPlanetmath of H (being contained in H and normal in G). By the First Isomorphism TheoremPlanetmathPlanetmath, G/K is isomorphic to a subgroup of Sp, and consequently |G/K|=|G:K| must p!; moreover, any divisor of |G:K| must also |G|=|G:K||K|, and because p is the smallest divisor of |G| different from 1, the only possibilities are |G:K|=p or |G:K|=1. But |G:K|=|G:H||H:K|=p|H:K|p, which |G:K|=p, and consequently |H:K|=1, so that H=K, from which it follows that H is normal in G. ∎

Title normality of subgroups of prime index
Canonical name NormalityOfSubgroupsOfPrimeIndex
Date of creation 2013-03-22 17:26:38
Last modified on 2013-03-22 17:26:38
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 13
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 20A05
Related topic Coset
Related topic GroupAction
Related topic ASubgroupOfIndex2IsNormal