Let be a finite group. Let denote the number of p-sylow subgroups. Prove that if then there exists p-sylow subgroups P and Q of G such that
if you take a look at the proof of Sylow theorems, you'll see that it is proved that if is a p-subgroup (Sylow or non-Sylow) and if are the p-Sylow subgroups of , then for some the result now follows easily from the given condition