Zero-separating invariants for finite groups

We fix a field kk of characteristic p. For a finite group G   denote by δ(G)δ(G) and σ(G)σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G   over kk and every v∈VG∖{0}v∈VG∖{0} or v∈V∖{0}v∈V∖{0} respectively, there exists a homogeneous invariant f∈k[V]Gf∈k[V]G of positive degree at most d   such that f(v)≠0f(v)≠0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisible by p  ). We show that δ(G)=|P|δ(G)=|P|. If NG(P)/PNG(P)/P is cyclic, we show σ(G)≥|NG(P)|σ(G)≥|NG(P)|. If G is p-nilpotent and P is not normal in G  , we show σ(G)≤|G|l, where l   is the smallest prime divisor of |G||G|. These results extend known results in the non-modular case to the modular case.