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

TL;DR

This paper investigates bounds on the p-part of degrees of irreducible p-Brauer characters and p-regular conjugacy class sizes in finite groups, establishing explicit inequalities relating these quantities.

Contribution

It provides explicit bounds connecting the p-part of Brauer character degrees with the p-part of the index of the p-core, and explores similar bounds for p-regular conjugacy class sizes.

Findings

Bound |G:O_p(G)|_p by p^{k * ē_p(G)} with an explicit constant k.

Establishes inequalities relating p-parts of character degrees and class sizes.

Analyzes the structure of finite groups via p-part bounds.

Abstract

Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ divides $\chi(1)$ for some $\chi \in IBr_p(G)$. We show that $|G:O_p(G)|_p \leq p^{k \bar e_p(G)}$ 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.