Simple abelian implies cyclic of prime order

Statement
Any simple Abelian group is a cyclic group of prime order.

Proof
The proof rests on two observations:


 * Subgroup of Abelian group implies normal: In an Abelian group, every subgroup is normal. Hence a simple Abelian group must have no proper nontrivial subgroup.
 * No proper nontrivial subgroup implies cyclic of prime order