Let be a finite group. Prove that is nilpotent.
We will use Frattini’s Argument to show that every Sylow subgroup is normal.
Let be a Sylow subgroup. By Frattini’s Argument, . By a lemma to a previous theorem, cannot be proper, so that and we have normal. Then is normal; thus all of the Sylow subgroups of are normal, and thus is nilpotent.