Sylow Subgroups

Quote:

Originally Posted by Chandru1

Let $G$ be a finite group. Let $n_{p}$ denote the number of p-sylow subgroups. Prove that if $n_{p} \ncong 1 \ (mod \ p^{2} )$ then there exists p-sylow subgroups P and Q of G such that $|P:P \cap Q|=|Q:Q \cap P|=p$

if you take a look at the proof of Sylow theorems, you'll see that it is proved that if $Q$ is a p-subgroup (Sylow or non-Sylow) and if $P_1, \cdots , P_k, \ \ k=n_p,$ are the p-Sylow subgroups of $G$ , then
$\sum_{i=1}^s |Q : P_i \cap Q| = k,$ for some $s \leq k.$ the result now follows easily from the given condition $k \ncong 1 \mod p^2.$