Degree of reductivity of a modular representation

For a finite-dimensional representation V of a group G over a field F, the degree of reductivity delta( G, V) is the smallest degree d such that every nonzero fixed point v epsilon V-G \{0} can be separated from zero by a homogeneous invariant of degree at most d. We compute delta( G, V) explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian p-groups.