Cyclic automorphism group not implies cyclic

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., aut-cyclic group) need not satisfy the second group property (i.e., cyclic group)
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Statement

It is possible to have a group that is an aut-cyclic group (i.e., the automorphism group is a cyclic group) but the group itself is not a cyclic group.

Related facts

• Finite and aut-cyclic implies cyclic
• Aut-cyclic implies abelian
• Finite abelian and aut-abelian implies cyclic
• Abelian and aut-abelian not implies locally cyclic

Proof

Let $G$ be the subgroup of the additive group of rational numbers comprising those rational numbers that, when written in reduced form, have denominators that are square-free numbers, i.e., there is no prime number $p$ for which $p^2$ divides the denominator. Then:

• The only non-identity automorphism of $G$ is the negation map, so the automorphism group is cyclic group:Z2, and is hence cyclic.
• $G$ is not a cyclic group. In fact, it is not even a finitely generated group because any finite subset of $G$ can only cover finitely many primes in their denominators. It is, however, a locally cyclic group: any finitely generated subgroup is cyclic.