Separating Invariants for Modular P -groups and Groups Acting Diagonally

We study separating algebras for rings of invariants of finite groups. We describe a separating subalgebra for invariants of p-groups in characteristic p using only transfers and norms. Also we give an explicit construction of a separating set for invariants of groups acting diagonally. Let F be an algebraically closed field and let G be a finite group. Consider a faithful representation ρ : G ↪→ GL(n,F) of degree n. It induces an action of the group G on the symmetric algebra on the dual space V ∗, which we denote by F[V ]. The subring of G-invariants is denoted by F[V ]. We note that the vector space V decomposes into disjoint G-orbits. We denote the orbit space by V/G = {[v] = {gv|g ∈ G}|v ∈ V }. Any invariant f ∈ F[V ] is constant on the G-orbits [v]. Indeed, F[V ] ⊆ F[V ] is the largest subalgebra with this property. A finitely generated graded subalgebra A ⊆ F[V ] (or more generally a subset in F[V ]) is called separating if for any two distinct G-orbits [v] 6= [w] there exists a function f ∈ A separating the two, i.e.,