On p-parts of Brauer character degrees and p-regular conjugacy class sizes

Christine Bessenrodt; Yong Yang

paper

On p-parts of Brauer character degrees and p-regular conjugacy class sizes

arXiv:1804.08241

Abstract

Let $G$ be a finite group, $p$ a prime, and $I B r_{p} (G)$ the set of irreducible $p$ -Brauer characters of $G$ . Let $e ˉ_{p} (G)$ be the largest integer such that $p^{e ˉ_{p} (G)}$ divides $χ (1)$ for some $χ \in I B r_{p} (G)$ . We show that $∣ G : O_{p} (G) ∣_{p} \leq p^{k e ˉ_{p} (G)}$ for an explicitly given constant $k$ . We also study the analogous problem for the $p$ -parts of the conjugacy class sizes of $p$ -regular elements of finite groups.