A sufficient condition for Sylow intersections of bounded index

Prove that if n_p \not\equiv 1 mod p^k, then there are distinct Sylow p-subgroups P and Q in G such that P \cap Q has index at most p^{k-1} in both P and Q.


(We will follow the strategy of this previous exercise.)

Let P \leq G be a Sylow p-subgroup. Now P acts on \mathsf{Syl}_p(G) by conjugation; note that if Q \in \mathsf{Syl}_p(G) is distinct from P, then using the Orbit-Stabilizer theorem and Lemma 4.19, we have |P \cdot Q| = [P : N_P(Q)] = [P : P \cap N_G(Q)] = [P : Q \cap P]. Since the orbit containing P itself has order 1, then n_p(G) = 1 + \sum [P : Q_i \cap P], where Q_i ranges over a set of orbit representatives. If every [P : Q_i \cap P] is divisible by p^k, then n_p(G) \equiv 1 mod p^k, a contradiction. Thus some [P : Q_i \cap P] is a power of p and is at most p^{k-1}.

We can see then that Q_i \cap P has index at most p^{k-1} in both P and Q_i.

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