Let be a Sylow -subgroup of and let be any subgroup of which contains . Prove that .
We prove a slightly stronger result by induction. First a definition.
Let be a finite group and a subgroup. The distance from to is the length of the longest chain of proper subgroups such that .
Lemma: Let be a finite group and a Sylow -subgroup. If are subgroups such that , then . Proof: We proceed by induction on the distance from to . For the base case, if , then so that , and we have . For the inductive step, suppose that for some , if the distance from to is at most then for all such that we have . Now let be a subgroup containing such that the distance from to is , and suppose further that . First, if , we have so that the conclusion holds. Suppose now that the inclusion is proper. Note that . By the induction hypothesis, we have . Recall that , so that by Sylow's Theorem. Modulo , we thus have .
In particular, if and , we have .