# Cyclic automorphism group implies abelian

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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## Contents

## Statement

Suppose is a group that is an Aut-cyclic group (?): the Automorphism group (?) is a Cyclic group (?). Then, is an Abelian group (?).

## Related facts

- Finite and cyclic automorphism group implies cyclic
- Classification of finite groups with cyclic automorphism group
- Cyclic automorphism group not implies cyclic
- Finite abelian and abelian automorphism group implies cyclic

## Facts used

- Group acts as automorphisms by conjugation
- Cyclicity is subgroup-closed
- Cyclic over central implies abelian

## Proof

**Given**: A group . The automorphism group is cyclic.

**To prove**: is abelian.

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let be the center of . Then, is isomorphic to a subgroup of , called the inner automorphism group . | Fact (1) | [SHOW MORE] | ||

2 | is a cyclic group. | Fact (2) | is cyclic | Step (1) | Step-fact-given combination direct. |

3 | is abelian | Fact (3) | Steps (1), (2) | Fact-step combination direct. |