# On the Group of order $pq$ where $p , q$ are primes .

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.

• Read and understand Theorem 18, page 139, in D&F – DonAntonio Feb 25 '13 at 0:23

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