# Finite group whose outer automorphism group is cyclic

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and group whose outer automorphism group is cyclic

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## Definition

A **finite group whose outer automorphism group is cyclic** is a finite group that is also a group whose outer automorphism group is cyclic, i.e., the outer automorphism group is a cyclic group (and in particular a finite cyclic group).

## Relation with other properties

### Weaker properties

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

Finite group with cyclic quotient of automorphism group by class-preserving automorphism group | ||||

Finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations | ||||

Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group |