This question is from Algebra by Thomas Hungerford , Section : Normal and Subnormal Series.
Prove that any group of order $p^2q$ (p,q primes) is solvable.
A subnormal series of a group G is a chain of subgroups $G= G_0>G_1 >...>G_n=<e>$. The subnormal series is Solvable if each factor $G_{i+1}/ G_{i}$ is abelian.
One result is : A finite group G is solvable if and only if G has a composition series whose factors are cyclic of prime order.
It has cyclic group of order p and q. Also, there exists a sylow p -subgroup of order $p^2$ , by sylow 3rd theorem. But only two groups of order $p^2$ are $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p \times \mathbb{Z}_p$. But $\mathbb{Z}_p \times \mathbb{Z}_p$ is not cyclic.
So, I think this result will not prove it.
I am unable to think of any other result. Can you please help?