# A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?

• In case you didn't see it: the accepted answer is wrong! Can you maybe un-accept it? Thanks! – j.p. Feb 24 '15 at 7:34
• Thanks for make me notice that. Unfortunately I know little about elliptic curves so I made the mistake of accepting without checking the correctness. I will ask this question also on Mathoverflow. – W4cc0 Feb 24 '15 at 8:28

Let $E$ be an elliptic curve over the complex numbers (to make things easy). $E$ is an abelian group which is not locally cyclic. For example, it contains the group $\mathbb Z \oplus \mathbb Z$ which is not locally cyclic.
Then it can be shown that if the j-invariant is neither $0$ or $1728$, then $Aut(E)\simeq \mathbb Z_2$, which is cyclic. The reference is this article.
• A hint in case you don't find any other automorphism of $(\mathbb{R}/\mathbb{Z})^2$: Look for vector space automorphisms of $\mathbb{R}$ seen as vector space over $\mathbb{Q}$ fixing $\mathbb{Q}\le\mathbb{R}$. – j.p. Feb 23 '15 at 10:57