Restrictions of characters in p-solvable groups

Let Gbe a p-solvable group, P <= Ga p-subgroup and chi is an element of Irr(G) such that chi(1)(p) >=vertical bar G : P vertical bar(p). We prove that the restriction chi P is a sum of characters induced from subgroups Q = Psuch that chi(1)(p)=vertical bar G : Q vertical bar(p). This generalizes previous results by Giannelli-Navarro and Giannelli-Sambale on the number of linear constituents of chi P. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer-Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde. (C) 2021 Elsevier Inc. All rights reserved.