Large orbits of nilpotent subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G. Then there exists v∈V such that |CH(v)|≤(|H|/p)1/p, where CH(v) is the centralizer of v in H and p is the smallest prime divisor of |H|.