Abelian 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., group whose automorphism group is abelian) need not satisfy the second group property (i.e., cyclic group)
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Statement

It is possible to have a group whose automorphism group is abelian but such that the group is not a cyclic group.

Related facts

Opposite facts

• Finite abelian and abelian automorphism group implies cyclic

Facts used

1. Locally cyclic implies abelian automorphism group, locally cyclic not implies cyclic
2. Abelian and abelian automorphism group not implies locally cyclic
3. Abelian automorphism group not implies abelian, cyclic implies abelian

Proof

We can counterexamples generated from Fact (1), (2), or (3).