Corollary

Every commutative simple group is cyclic of prime order.

Let G be such a group. Then $G \neq \{e\}$; let $a \neq e$ be an element of G. The subgroup generated by a is normal since G is commutative, it is not reduced to $\{e\}$ and hence is equal to G. Therefore G is monogenous and hence isomorphic to a group of the form $\mathbf{Z}/p\mathbf{Z}$ with $p > 0$, since $\mathbf{Z}$ is not simple, and $p$ is necessarily prime.