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


This question was first posted [here](https://math.stackexchange.com/questions/1160453/a-group-whose-automorphism-group-is-cyclic?noredirect=1#comment2370809_1160453).



A group whose automorphism group is cyclic
There's a construction of a rank two (and therefore not locally cyclic) abelian group with endomorphism ring $\mathbb{Z}$, and therefore automorphism group cyclic of order 2, in "On the cancellation of modules in direct sums over Dedekind domains" by L. Fuchs and F. Loonstra,
Indagationes Mathematicae, Volume 74, (1971), 163-169
([link](http://www.sciencedirect.com/science/article/pii/S1385725871800227))


