# Group whose outer automorphism group is cyclic

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

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

1. The outer automorphism group of the group is a cyclic group.
2. 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