Prove that if mod , then there are distinct Sylow -subgroups and in such that has index at most in both and .
(We will follow the strategy of this previous exercise.)
Let be a Sylow -subgroup. Now acts on by conjugation; note that if is distinct from , then using the Orbit-Stabilizer theorem and Lemma 4.19, we have . Since the orbit containing itself has order 1, then , where ranges over a set of orbit representatives. If every is divisible by , then mod , a contradiction. Thus some is a power of and is at most .
We can see then that has index at most in both and .