I'm working on the following question while studying for my qual. in Algebra: Suppose that $G$ is a finite group with $p+1$ Sylow $p$-subgroups. Show that if $P$ and $Q$ are distinct Sylow $p$-subgroups of $G$, then their intersection is a normal subgroup of $G$

There was a hint to show that $|P : P \cap Q| = p$, which I've established, but I'm unsure of how this helps. Do anyone have any suggestions?


An index $p$ subgroup of a finite $p$-group is always normal, so $P\cap Q$ is normal in $P$, and in $Q$....

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    $\begingroup$ Fixed under conjugation. Thank you! $\endgroup$ – mathgenesis22813 Jul 2 '17 at 16:23

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