Prime Group is Simple

Theorem

Groups of prime order are simple.

Proof

Follows directly from Prime Group has no Proper Subgroups: a group of prime order has only itself and the trivial group as subgroups.

From Trivial Subgroup and Group Itself are Normal, these subgroups are normal.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups: Example $123$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 59 \epsilon$