On p-parts of character degrees

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

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