Prove that a group of order $pq$ has a unique Sylow $q$-subgroup and deduce that it is solvable.

Question

Suppose $p$ and $q$ are prime numbers with EDIT: $q>p$. Prove that a group of order $pq$ has a unique Sylow $q$-subgroup and deduce that it is solvable.

I've found this question in an old past paper and I'm struggling to find the answer and method. Any help/solutions appreciated!

Answer 1

I think you have something a bit backwards. There should be a unique Sylow $p$-subgroup, not $q$-subgroup. By the Sylow theorems, if we let $n_p$ be the number of $p$-Sylow subgroups, then we have $$n_p \equiv 1 \pmod{p}.$$

But we also have $n_p | q < p$ implying that $n_p = 1$.

EDIT: This is under the assumption that $p$ is the bigger prime.