# proof of Frattini argument

Let $g\in G$ be any element. Since $H$ is normal, $gSg^{-1}\subset H$. Since $S$ is a Sylow subgroup of $H$, $gSg^{-1}=hSh^{-1}$ 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 ProofOfFrattiniArgument 2013-03-22 13:16:05 2013-03-22 13:16:05 bwebste (988) bwebste (988) 4 bwebste (988) Proof msc 20D20