Finite abelian and abelian automorphism group implies cyclic

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).

Related facts

Similar facts

• Cyclic implies abelian automorphism group
• Abelian automorphism group implies class two
• Abelian automorphism group not implies abelian
• Cyclic automorphism group implies abelian
• Finite and cyclic automorphism group implies cyclic
• Classification of finite groups with cyclic automorphism group

Opposite facts

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

Proof

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