Finite abelian and abelian automorphism group implies cyclic

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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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Statement

Suppose G is a finite abelian group that is also a group whose automorphism group is abelian, i.e., the automorphism group \operatorname{Aut}(G) is also a (finite) abelian group. Then, G is a cyclic group (and hence, a finite cyclic group).

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Proof

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