Large -orbits for -nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors and in V such that . A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Brou, and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.