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

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

Suppose is a finite set, is the set of conjugacy classes of , and is the set of equivalence classes of irreducible representations of over . The automorphism group acts on the sets as well as . We say that is a **finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations** if the permutation representations of on and are equivalent, i.e., and are equivalent as -sets.

## 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 | ||||

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

Finite cyclic group |

### Weaker properties

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

Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group | same orbit sizes of conjugacy classes and irreducible representations under automorphism group does not imply that automorphism group has equivalent actions |