On the largest Brauer and p'-character degrees

Let K be a field of characteristic p > 0. We prove that if all irreducible representations over K of a finite group G have degree at most n, then G has a characteristic subgroup N of index bounded above in terms of n such that its derived subgroup N' is a p-group. Our proof of this result relies on the celebrated Larsen-Pink theorem. We also show that every finite group G has a solvable p-nilpotent subgroup of index bounded above in terms of the largest p1-degree of the complex irreducible characters of G.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).