proof of Frattini argumentLet g∈G be any element. Since H is normal, gSg-1⊂H.Since S is a Sylow subgroup of H, gSg-1=hSh-1 for some h∈H, by Sylow’s theorems.Thus n=h-1g normalizes S, and so g=hn for h∈H and n∈NG(S).