On p-parts of character degrees and conjugacy class sizes of finite groups

TL;DR

This paper proves a bound relating the p-part of the index of the Fitting subgroup in a finite group to the largest p-part of character degrees, confirming a conjecture by Moretó and exploring conjugacy class sizes.

Contribution

It establishes a new inequality connecting p-parts of group structure and character degrees, settling a conjecture and analyzing conjugacy class sizes.

Findings

Proved that |G:F(G)|_p ≤ p^{k e_p(G)} for a constant k.

Confirmed a conjecture of A. Moretó regarding p-parts of character degrees.

Explored the relationship between conjugacy class sizes and p-parts in finite groups.

Abstract

Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $\chi(1)$ for some $\chi \in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k e_p(G)}$ for a constant $k$. This settles a conjecture of A. Moret\'o. We also study the related problems of the $p$-parts of conjugacy class sizes of finite groups.