On p-parts of character degrees

TL;DR

This paper establishes sharp bounds on the p-part of the quotient of a finite group by its p-core, under specific conditions on character degrees, thereby resolving two conjectures in the field.

Contribution

It provides the first definitive bounds on |G/O_p(G)|_p under conditions on irreducible character degrees, settling two longstanding conjectures.

Findings

Sharp bounds for |G/O_p(G)|_p established

Conditions on character degrees are crucial for the bounds

Conjectures by Lewis, Navarro, Tiep, and Tong-Viet are confirmed

Abstract

In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption that either $p^2 \nmid \chi(1)$ for all $\chi \in {\rm Irr}(G)$ or $p^2 \nmid \phi(1)$ for all $\phi \in {\rm IBr}_p(G)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet.