Question: if $G$ is non-abelian group of order $pq$ where $p$ and $q$ are distinct primes such that, $p<q$ and $p$ does not divides $q-1$ then, is $G$ is unique? If yes then, how it's looks?
My attempt: yes, $G$ must be unique and, it is nothing but, $\mathbb{Z}_q⋊ \mathbb{Z}_p $ that is, semi direct product of $\mathbb{Z}_q$ and $\mathbb{Z}_p$. Is am I right?
But I didn't know how to prove this semidirect product is unique. Please help me.
