Can anyone give me an example of:
 An infinite abelian but noncyclic group whose automorphism group is cyclic.
Can anyone give me an example of:



Let $G$ be the (nonfinitely generated) additive group of rational numbers with squarefree denominators. Then the only automorphism of $G$ is negation. From here. 


One has a very general statement in this direction: For any cotorsionfree ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$ In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$modules, where $R$ is any commutative, cotorsionfree ring. Then you obtain $A$  any cotorsionfree $R$algebra. You will find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012). cotorsionfree above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Z}^\wedge_p\to A$ for all primes $p$. 

