Zeros of Brauer characters and linear actions of finite groups

Let G be a finite group, and p a prime number greater than 3. It is known that, if every irreducible p-Brauer character of G does not vanish on any p′-element of G, then G is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, we show that the p′-length of G is at most 2 (the bound being the best possible). The structural results are obtained as an application of the main theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces.