Let be a finite group. Use the method of proof in Sylow’s Theorem to show that if mod , then there exist distinct Sylow -subgroups and such that .
Let be the Sylow -subgroups of .
Now let be any -subgroup of . Note that acts on by conjugation; we can then say . Relabel the elements of so that for are a set of orbit representatives of the action of . Note that by the Orbit-Stabilizer Theorem, so that by Lemma 19 in the text. Thus , where ranges over a set of orbit representatives of the action of on .
We may let itself be a Sylow -subgroup, and (without loss of generality and with a relabeling if needed) choose . Then . Now because is a -subgroup, for some . Modulo , then, . If (mod ) for all , then we have , a contradiction. Thus there exists a Sylow -subgroup such that .
Finally, note that , by Lagrange and since .