proof of Frattini argument

Let $g\in G$ be any element. Since $H$ is normal, $gS{g}^{-1}\subset H$. Since $S$ is a Sylow subgroup of $H$, $gS{g}^{-1}=hS{h}^{-1}$ for some $h\in H$, by Sylow’s theorems. Thus $n=h^{-1}g$ normalizes $S$, and so $g=hn$ for $h\in H$ and $n\in N_G(S)$.

Title	proof of Frattini argument
Canonical name	ProofOfFrattiniArgument
Date of creation	2013-03-22 13:16:05
Last modified on	2013-03-22 13:16:05
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	4
Author	bwebste (988)
Entry type	Proof
Classification	msc 20D20