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: $\S 59 \epsilon$
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