On p-parts of character degrees and conjugacy class sizes of 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 $χ(1)$ for some $χ\in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k e_p(G)}$ for a constant $k$. This settles a conjecture of A. Moretó. We also study the related problems of the $p$-parts of conjugacy class sizes of finite groups.

arXiv admin note: substantial text overlap with arXiv:1501.03237