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15.62 (2002)

Solved

Given an ordinary irreducible character $\chi$ of a finite group $G$ write $p^{e_p(\chi)}$ to denote the $p$-part of $\chi(1)$ and put
$$e_p(G) = \max \{ e_p(\chi) \mid \chi \in \text{Irr}(G) \}.$$ Suppose that $P$ is a Sylow $p$-subgroup of a group $G$. Is it true that $e_p(P)$ is bounded above by a function of $e_p(G)$?

Contributor: A. Moretó

Progress

Comment of 2005: An affirmative answer in the case of solvable groups has been given in (A. Moretó, T. R. Wolf, Adv. Math., 184 (2004), 18–36).
Yes, it is true (Yong Yang, Guohua Qian, Adv. Math., 328 (2018), 356–366).

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