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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 59 \zeta$