group of order $pq^n$ is solvable?

Question

If $p<q$ be primes and $n \geq 0$ integer. If $G$ is a group such that its order is $pq^n$, then $G$ is solvable.

I tried to use Sylows theorem to find number of Sylows subgroups but I stuck with finding $n_q$. Is my strategy right or need to change it?

Thanks

Answer 1

By Burnside's theorem any group of order $p^kq^m$ where $p,q$ are prime is solvable https://en.m.wikipedia.org/wiki/Burnside%27s_theorem

Without the theorem. The number of Sylow $q$-subgroups should divide $p$ and is $\equiv 1\mod q$. Since $p<q$ it is only possible if there is one Sylow $q$-subgroup which is normal and nilpotent. The quotient group is cyclic of order $p$ so the group is solvable.