## The normalizer of a maximal Sylow intersection is not a p-subgroup

Suppose that over all pairs of Sylow $p$-subgroups, $P$ and $Q$ are chosen so that $|P \cap Q|$ is maximal. Prove that $N_G(P \cap Q)$ is not a $p$-group.

We proved this in the course of answering this previous exercise.

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