# Finite abelian and abelian automorphism group implies cyclic

From Groupprops

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 is a finite abelian group that is also a group whose automorphism group is abelian, i.e., the automorphism group is also a (finite) abelian group. Then, 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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