Classification of groups of order a product of two distinct primes

Statement
Suppose $$p$$ and $$q$$ are distinct prime numbers with $$p < q$$. Then, there are two possibilities for the number of isomorphism classes of groups of order $$pq$$:


 * 1) If $$p$$ does not divide $$q - 1$$, then there is only one isomorphism class of groups of order $$pq$$, namely, the cyclic group.
 * 2) If $$p$$ divides $$q - 1$$, then there are two possibilities: the cyclic group of order $$pq$$ and the semidirect product $$\mathbb{Z}_q \rtimes \mathbb{Z}_p$$ where $$\mathbb{Z}_q$$ is thought of as the additive group of integers mod $$q$$ and $$\mathbb{Z}_p$$ is identified with the subgroup of order $$p$$ in $$\mathbb{Z}_q^\ast$$, which is cyclic of order $$q - 1$$.

Related facts

 * Every Sylow subgroup is cyclic implies metacyclic
 * Square-free implies solvability-forcing

Facts used

 * 1) uses::Every Sylow subgroup is cyclic implies metacyclic