Original (Flawed) Question
Why is it that if a group is of the order $pqn$ where $p, q$ are distinct primes and $n$ is some integer coprime to $p$ and $q$, then there is a non-abelian subgroup of order $pq$? (I am reading some notes and the author says this without proving it, so I assume it is very elementary.)
Sorry about this confusing question, I think I have misunderstood it. (As @QiaoChuYuan kindly suggested.) It should be saying for a cyclic group of order $pqn$ where $p,q,n$ are as described above, AND $p$ divides $q-1$ then there is a non-abelian subgroup of order $pq$. Does this make sense now? If so could someone please tell me why it is true? Thank you.