Generators of a Group of order $pq$

Question

I can't really figure out how to solve this exercise: Let $G$ be a cyclic group of order $pq$ (two primes) with $H$ a (normal) subgroup such that $\frac GH$ is also cyclic -by the way isn't that always the case?- prove that $G$ has two generators. I tried to enstablish some isomorphism beetween $G$ and $\mathbb{Z}_{pq}$ but I'm not sure how to proceed and use the hypothesis that $\frac GH$ is also cyclic. Another thing that bugs me is that I can't understand if I have to prove $G$ has only two generators or at least two of them.

Answer 1

Every cyclic group of order $>2$ has at least two generators: if $x$ is one generator $x^{-1}$ is another one. (And they are distinct as $\operatorname{ord}(x) \neq 2$)

In general $\bar m$ is a generator of $\mathbb{Z}/n\mathbb{Z}$ iff $\gcd(m,n)=1$. This also shows that there can be more than $2$ generators for cyclic groups of order $pq$, for example if $p=2, q=5$, then there are 4 generators in $\mathbb{Z}/10\mathbb{Z}$: $\bar{1},\bar{3},\bar{7},\bar{9}$.