On p-parts of character degrees

Abstract

Let G be a finite group and p be a prime. In this paper, we get the sharp bound for $|G/O_p(G){|}_p$ under the assumption that either $p^2\nmid \chi \left(1\right)$ for all $\chi \in \mathrm{Irr}\left(G\right)$ or $p^2\nmid \varphi \left(1\right)$ for all $\varphi \in {\mathrm{IBr}}_p\left(G\right)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet in [7].