Let G be a finite group and P be a cyclic Sylow p-subgroup of order p and generator x, where p is the smallest prime that divides |G|. Show that $|N_G(P) : C_H(x)| < p$.
As a hint, I proved that $C_G(P) = C_H(P) = C_H(x)$, where $H = N_G(P)$. I am trying to bound the index via $P \le C_H(P) \le N_H(P) \le H \le G$ but can't figure out the proof. Any help?