Finite simple unisingular groups of Lie type

Let G be a finite simple group of Lie type in characteristic p. We investigate how large the set of p′-elements acting without fixed points on an irreducible G-module in characteristic p can be. This comes up naturally in studying derangements for finite primitive groups and also in a computational group algorithm proposed by Babai and Shalev. We also consider the same problem for algebraic groups.