After seeing that groups of order $pq$ are never simple, we now study their structure more precisely. Let $p$ and $q$ be primes with $p>q$. The Sylow $p$-subgroup is forced to be normal, and the remaining question is how a subgroup of order $q$ acts on it by conjugation. In this lesson, we focus in the groups of order pq. The main result says that such a group is either cyclic or a nonabelian semidirect product, and the nonabelian case can occur only when $q$ divides $p-1$.

Let $p$ and $q$ be prime integers with $p>q$. A \textbf{group of order pq} is a finite group $G$ such that $|G|=pq$.