On p-parts of character degrees

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 $\chi \in \mathrm{Irr}(G)$ or $p^2\nmid \varphi (1)$ for all $\varphi \in {\mathrm{IBr}}_p(G)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet.