A result on character degrees and conjugacy class sizes in finite groups

Abstract

Let G be a finite group and π be a set of primes including at least two elements. We write cd(G) and cs(G) to denote the set of complex irreducible character degrees and conjugacy class sizes of G, respectively, and write π(m) to denote the set of all prime divisors of a positive integer m. For any 1 ≠ m ∈ (G) and 1 ≠ m ∈ (G), in this note, we shall present the corresponding group structures of finite group G in the case π(m) = π, respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.