# 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).

## Proof

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