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