\section{Applications of Sylow Theorems}

\begin{theorem}[Cyclic Groups of Order $\mathbf{pq}$]
	If $G$ is a group of order $pq$, where $p$ and $q$ are primes, $p < q$, and $p$ does not divide $q - 1$, then $G$ is cyclic. In particular, $G$ is isomorphic to $\Z_{pq}$.
\end{theorem}