# Group whose outer automorphism group is cyclic

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group whose outer automorphism group is cyclic** is defined in the following equivalent ways:

- The outer automorphism group of the group is a cyclic group.
- The inner automorphism group is a 1-completed subgroup of the automorphism group, i.e., the automorphism group can be generated by the inner automorphism group and one element.

## Relation with other properties

### Conjunction with other properties

We are often interested in this property in the context of finite groups: finite group whose outer automorphism group is cyclic.

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Group in which every automorphism is inner | outer automorphism group is trivial | |||

Complete group | centerless, every automorphism is inner | |||

Group whose inner automorphism group is maximal in automorphism group |