Cyclic Group is Simple iff Prime

Theorem

Let $G$ be a cyclic group.

Then $G$ is simple if and only if $G$ is a prime group.

Proof

Let $G$ be a cyclic group.

From Cyclic Group is Abelian it follows that $G$ is an abelian group.

The result follows from Abelian Group is Simple iff Prime.

$\blacksquare$