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

W4cc0
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    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 07:34
  • 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 08:28

2 Answers2

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Warning: This answer is not correct (see comments)!

Yes.

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.

hardmath
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Fredrik Meyer
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    I'd read the original question as speaking of automorphisms as an abelian group. In your example the two automorphisms are the only ones that additionally preserve the structure as an abelian variety. There are surely more group automorphisms that do not preserve this structure. – j.p. Feb 23 '15 at 08:20
  • 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
  • @j.p. Good point! It seems there is no way to rescue my argument. Do you have any ideas on the truthness of the question? – Fredrik Meyer Feb 23 '15 at 14:21
  • I thought about it, but couldn't come up with alternatives (I tried variations of Jack Smith's example for an older question by W4cco). I tend to believe that the answer is no, and that the question could be asked at mathoverflow.net (but infinite abelian groups ain't my strength). – j.p. Feb 23 '15 at 15:05
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This MathOverflow question cites this paper, which says there are torsion-free groups of any finite rank of without automorphism group $C_2$, and which in turn cites this paper in which Theorem III says there are Abelian groups of all finite ranks $\ge 3$ with automorphism group $C_2$. Rank $>1$ means they can't be locally cyclic. (the paper refers to an automorphism group $C_2$ as having only 'trivial' automorphisms.) So the answer is yes.

Zoe Allen
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