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., aut-abelian group)
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Statement

Any locally cyclic group is an aut-abelian group, i.e., it has an abelian automorphism group.

Related facts

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

Proof

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