Let $P$ be a finite group admitting a fixed-point-free automorphism of prime order. Then $P$ is nilpotent. The proof is Thompson's 1959 doctoral thesis; it uses deep results in group theory well beyond the character-theoretic methods of this course. Applied to the Frobenius kernel $K$ — which admits fixed-point-free automorphisms from $H$ — Thompson's theorem implies that $K$ is nilpotent, hence a direct product of its Sylow subgroups.