Show that a group is soluble I am wondering how I would go about showing that any group of order $p^2q$ or $pq^2$ is soluble, where $p,q$ are primes with $p<q$.
Could you give me a hint or outline for how to get started please? Thanks!

Hint : Let $G$ be your group of order $p^2q$. Consider a $p$-Sylow subgroup $P$ and a $q$-Sylow subgroup $Q$ of $G$. You have to show that either $P$ or $Q$ is normal in $G$, in which case you'll have a normal series for $G$ ($p$-groups are solvable). To do this, try computing the number of $p$-Sylow subgroups and number of $q$-Sylow subgroups. If either of the one is $1$, then normality follows and you'll have a solution.
