I am classifying the groups of order $pq$, where $p$ and $q$ are primes ($p<q$) and $p$ divides $q-1$.
Let $H\cong \mathbb{Z}/q\mathbb{Z}$ be the only q-Sylow subgroup of $G$ and $K\cong \mathbb{Z}/p\mathbb{Z}$ a p-Sylow subgroup of $G$. Then, $G\cong H \rtimes_{\varphi} K$, with
\begin{align} \varphi: \mathbb{Z}/p\mathbb{Z} &\to Aut(\mathbb{Z}/q\mathbb{Z})\\ b&\mapsto \varphi_b: \mathbb{Z}/q\mathbb{Z} \to \mathbb{Z}/q\mathbb{Z} \end{align}
I already know that $|\varphi(\mathbb{Z}/p\mathbb{Z})|=p$ since $$\varphi(\mathbb{Z}/p\mathbb{Z}) \leq Aut(\mathbb{Z}/q\mathbb{Z}),\ \quad \quad p=|(\mathbb{Z}/p\mathbb{Z})| \text{ divides } |\varphi(\mathbb{Z}/p\mathbb{Z})|$$
So $\varphi$ is not trivial. However, I don't know how this implies that $G$ is not abelian. Any help would be appreciated.
