Group of Order p^2 q is not Simple
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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: $\S 59 \zeta$