Group of Order 30 is not Simple
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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: $\S 59 \epsilon$