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

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **finite group with cyclic quotient of automorphism group by class-preserving automorphism group** is a finite group that is also a group with cyclic quotient of automorphism group by class-preserving automorphism group, i.e., a group such that the quotient group of its automorphism group by the normal subgroup comprising the class-preserving automorphisms is a cyclic group (and in particular, a finite cyclic group).

## Relation with other properties

### Stronger properties

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

Finite complete group | ||||

Finite group whose outer automorphism group is cyclic |

### Weaker properties

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

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 | Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group |