On degrees of irreducible Brauer characters

Abstract

Based on a large amount of examples, which we have checked so far, we conjecture that | G | p ′ ≤ ∑ φ φ ( 1 ) 2 |G|_{p’}\le \sum _\varphi \varphi (1)^2 where p p is a prime and the sum runs through the set of irreducible Brauer characters in characteristic p p of the finite group G G . We prove the conjecture simultaneously for p p -solvable groups and groups of Lie type in the defining characteristic. In non-defining characteristics we give asymptotically an affirmative answer in many cases.