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Groupprops β

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

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