Group of Order p^2 q is not Simple

Theorem
Let $p$ and $q$ be prime numbers such that $p \ne q$.

Let $G$ be a group of order $p^2 q$.

Then $G$ is not simple.

Proof
From Group of Order $p^2 q$ has Normal Sylow $p$-Subgroup, $G$ has a normal subgroup of order $p^2$.

Hence the result, by definition of simple group.