Groups of order pq

Example

Let $p < q$ be prime numbers such that $p$ does not divide $q - 1$, and let $G$ be a group of order $pq$. The number $n_q$ divides $p$ and satisfies $n_q \equiv 1 \pmod q$, so $n_q = 1$, because $p < q$. The number $n_p$ divides $q$ and satisfies $n_p \equiv 1 \pmod p$; since $p \nmid q - 1$, $n_p = 1$. So $G$ has a normal subgroup $P$ of order $p$ and a normal subgroup $Q$ of order $q$. They meet only in $e$, so $G$ is the internal direct product of $P$ and $Q$. Thus $G \cong \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} \cong \mathbb{Z}/pq\mathbb{Z}$, and $G$ is cyclic. For example, each group of order $15$ is cyclic. If $p$ divides $q - 1$, there is also a group of order $pq$ that is not cyclic, for example the dihedral group $D_3$ of order $6$.