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

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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## Definition

A finite group $G$ is said to have the same orbit sizes of conjugacy classes and irreducible representations under automorphism group if the following holds:

Let $C(G)$ be the set of conjugacy classes of $G$ and $R(G)$ be the set of irreducible representations (up to equivalence) of $G$ over $\mathbb{C}$. The automorphism group $\operatorname{Aut}(G)$ acts on both $C(G)$ and $R(G)$. The condition we need is that the orbit sizes in $C(G)$ and $R(G)$ be equal.

## Relation with other properties

### Stronger 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 same orbit sizes of conjugacy classes and irreducible representations under automorphism group does not imply that automorphism group has equivalent actions
Finite group whose outer automorphism group is cyclic
Finite group with cyclic quotient of automorphism group by class-preserving 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
Finite abelian group