# $Aut(G) \text{ is cyclic} \Rightarrow G \text{ is cyclic}$?

Let $G$ be a group. Is true that, if $Aut(G)$ be a cyclic group then, $G$ is cyclic?!

I know that $Aut(G) \text{ is cyclic} \Rightarrow G \text{ is Abelian}$, but above question asks something stronger!

The subgroup of the group of rational numbers $\mathbb{Q}$ generated be all the elements with prime denominator has an automorphism group which is isomorphic to $\mathbb{Z}_2$.

You may look at this question for further details.

Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2