Locally cyclic implies abelian automorphism group

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., locally cyclic group) must also satisfy the second group property (i.e., group whose automorphism group is abelian)
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Statement

Any locally cyclic group is a group whose automorphism group is abelian, i.e., it has an abelian automorphism group.

Related facts

• Cyclic implies abelian automorphism group
• Abelian and abelian automorphism group not implies locally cyclic
• Abelian automorphism group not implies abelian
• Finite abelian and abelian automorphism group implies cyclic

Proof

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