Let $\displaystyle p$ be a prime, $\displaystyle G$ a finite group, and $\displaystyle P$ a $\displaystyle p$-Sylow subgroup of $\displaystyle G$. Let $\displaystyle M$ be any subgroup of $\displaystyle G$ which contains $\displaystyle N_G(P)$. Prove that $\displaystyle [G:M]\equiv 1$ (mod $\displaystyle p$). (Hint: look carefully at Sylow's Theorems.)