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

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

Suppose $G$ is a finite set, $C(G)$ is the set of conjugacy classes of $G$, and $R(G)$ is the set of equivalence classes of irreducible representations of $G$ over $\mathbb{C}$. The automorphism group $\operatorname{Aut}(G)$ acts on the sets $C(G)$ as well as $R(G)$. We say that $G$ is a finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations if the permutation representations of $\operatorname{Aut}(G)$ on $C(G)$ and $R(G)$ are equivalent, i.e., $C(G)$ and $R(G)$ are equivalent as $\operatorname{Aut}(G)$-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