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

Question

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!

score 6 · Answer 1 · edited Apr 13 '17 at 12:21

6

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

edited Apr 13 '17 at 12:21

Community

1

answered Feb 22 '15 at 15:45

W4cc0

4,160

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

1 Answers1