Group of Order 30 is not Simple

Theorem

Let $G$ be a group of order $30$.

Then $G$ is not simple.

Proof

From Group of Order 30 has Normal Cyclic Subgroup of Order 15, $G$ has a normal subgroup of order $15$.

Hence the result, by definition of simple group.

$\blacksquare$

Sources

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