Prime divisors of irreducible character degrees

Abstract

Let $G$ be a finite group. We denote by $\rho(G)$ the set of primes which divide some character degrees of $G$ and by $\sigma(G)$ the largest number of distinct primes which divide a single character degree of $G$. We show that $|\rho(G)|\leq 2\sigma(G)+1$ when $G$ is an almost simple group. For arbitrary finite groups $G$, we show that $|\rho(G)|\leq 2\sigma(G)+1$ provided that $\sigma(G)\leq 2$.