Can anyone give me an example of:



> 
> * An infinite abelian but non-cyclic group whose automorphism group is cyclic.
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> 
> 



Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic
Let $G$ be the (non-finitely generated) additive group of rational numbers with square-free denominators. Then the only automorphism of $G$ is negation. 


From [here](https://groupprops.subwiki.org/wiki/Aut-cyclic_not_implies_cyclic).


