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Let $G$ be a group, $\lvert G \rvert = pq$ where $p$, $q$ are primes, $q$ is bigger than $p$.

Let $P$ be a Sylow $p$-subgroup and $Q$ be a Sylow $q$-subgroup and let $n_p$= the number of Sylow $p$-subgroups. Prove that $n_p$ divides $q$, $n_q$ divides $p$


This fact is used in proving some statements in Dummit and Foote, page 143, 3rd ed, but I don't know why is it true, so I post my question.

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    $\begingroup$ Read and understand Theorem 18, page 139, in D&F $\endgroup$ – DonAntonio Feb 25 '13 at 0:23
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Quick "answer", which must be obvious if you really read that theorem in page 143 in D&F, which is not other than Sylow's Theorem.

In this theorem's proof one sees that if $\,|G|=p^\alpha m\,\, ,\,\, p^{\alpha+1}\nmid|G|\,$ , then in fact, if $\,P\,$ is any Sylow $\,p-$Sylow subgroup of $\,G\,$ , we get

$$n_p=[G:N_G(P)]=1\pmod p\Longrightarrow n_p\mid m$$

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  • $\begingroup$ @amWhy , what does " leading a horse to water " mean ? $\endgroup$ – Fawzy Hegab Feb 26 '13 at 0:30
  • $\begingroup$ MrWhy: I was just hoping that the answer would provide the help you need for answering the question, that's all! $\endgroup$ – Namaste Feb 26 '13 at 1:01

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