Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.
For proof i employ the induction on $ \vert G \vert $. I should show $ \Phi(G) = 1 $. If $ \Phi(G) \neq 1 $, then $ P\Phi(G)/\Phi(G) $ is normal in $ G/\Phi(G) $, $ P\Phi(G) \unlhd G $. $ P\Phi(G) $ is nilpotent. But i can't tell contradiction.