# Finite and cyclic 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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## Contents

## Statement

Suppose is a finite group that is an Aut-cyclic group (?), i.e., the automorphism group of is a cyclic group. Then, is itself a Cyclic group (?), and in particular, a Finite cyclic group (?).

Note that the converse is *not* true, i.e., the automorphism group of a finite cyclic group need not be cyclic.

## Related facts

### Similar facts

- Trivial automorphism group implies trivial or cyclic of order two
- Finite abelian and aut-abelian implies cyclic
- Cyclic implies aut-abelian
- Aut-cyclic implies abelian
- Aut-abelian implies class at most two

### Opposite facts

- Aut-cyclic not implies cyclic
- Aut-abelian not implies abelian
- Abelian and aut-abelian not implies locally cyclic

## Facts used

## Proof

### Proof from given facts

**Given**: A finite group such that is cyclic.

**To prove**: is cyclic.

**Proof**

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

1 | is abelian. | Fact (1) | is cyclic. | -- | given + fact direct |

2 | is abelian. | Fact (2) | is cyclic. | -- | given + fact direct |

3 | is cyclic | Fact (3) | is finite | Steps (1), (2) | Fact + step + given direct. |