If G is abelian and simple, then G is cyclic.

Question

True/False: If a group $G$ is abelian and simple,then $G$ is cyclic.

Solution:

True. If $G$ is an abelian simple group, then $G$ is isomorphic to $\mathbb{Z}_p$ for some prime $p$.

Answer 1

Assume $G$ is abelian and simple. Then any subgroup is a normal subgroup, so the only subgroups must be $\{1\}$ and $\{G\}$ if $G$ is simple.

If $x\neq 1$ is an element of $G$, then $\langle x\rangle=\{x^k\mid k\in\mathbb Z\}$, the subgroup of $G$ generated by $x$, must be all of $G$. So $G$ is cyclic.