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Let $p > q$ be two primes such that $q | (p-1)$ . Prove that there exists a non-abelian group of order $pq$

This is not a homework problem. We were asked to prove this in a final exam and I did not mange to solve it and I'd like to know the answer anyway.

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    There are two ways that come to mind: Have you seen semidirect products? If not, one can consider affine maps from a finite field to itself. – 2017-01-04
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    @TobiasKildetoft, I've seen them before (many years ago via self-study but I forgot them entirely) but they were not covered in class, so I suppose the answer should not use them – 2017-01-04

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$\Bbb{Z}_p^\times$ is a group of order $p-1$. Since $q$ divides $|\Bbb{Z}_p^\times|$, by Cauchy's Theorem, the group contains a subgroup $X$ of order $q$.

Let $G=\{\begin{pmatrix}x&&0\\z &&1\end{pmatrix}| \; x\in X,z\in \Bbb{Z_p}\}$ be a subset of $GL(2,\Bbb{Z}_p)$.
Clearly $|G|=pq$.

Try to verify that $G$ is closed and non-abelian.