## In a finite group, if the number of Sylow subgroups is not 1 mod p² then there exist Sylow subgroups which intersect maximally

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 .

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