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 $n\in N_G(S)$.