Let $G$ be an [abelian group](/page/Abelian%20Group), written multiplicatively, with identity element $e$. Then $G$ is simple if and only if there exists a [prime number](/page/Prime%20Number) $p$ such that $G \cong C_p$, where $C_p$ denotes the [cyclic group](/page/Cyclic%20Group) of order $p$.