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I'm looking to prove that if $G$ is a group of order $pk$ where $p$ is prime and $p>k$, that any subgroup $K\leq G$ of order $p$ is normal in G.

Does anybody have any hints or tips for proving this? I've tried multiple approaches including induction on $k$, and haven't really gotten anywhere.

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This follows from the third Sylow theorem.

Consider how your subgroup of order $p$ acts on the whole group $G$ and what you can infer from the orbit-stabilizer theorem in this case.

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