Finite groups with only one p-singular Brauer character degree

Abstract

Recently, Isaacs, Moretó, Navarro, and Tiep investigated finite groups with just one irreducible character degree divisible by a given prime p, and showed that their Sylow p-subgroups are almost normal and almost abelian. In this paper, we consider the corresponding situation for Brauer characters. In particular, we show that if a finite group G has just one irreducible p-Brauer character degree n divisible by p≥5 then either G/Op(G) has a non-normal T.I. Sylow p-subgroup of order np, or G has a nonabelian chief factor of order divisible by p that is unique and is a simple group of Lie type of characteristic p.