Prove that G is a cyclic group 
Suppose that $|G| = pq$ where $p$ and $q$ are  primes such that $p < q$ and $p$ does not divide $q − 1$. Prove that $G$ is a cyclic group.

A cyclic group is a group that has a unique generator element, so is the way to go with this to find that element? Or am I missing something?

In my answer here I showed that any such group must be abelian. Now it is a general fact that an abelian group of order $pq$ where $p \ne q$ are primes, then the group is cyclic. The goal is to produce elements of order $p$ and $q$ and then find the order of the product. 
The advantage of this approach is that it can be done without Sylow Theorems. 
