On Generators of Modular Invariant Rings of Finite Groups

Let G be a finite group, let V be an FG-module of finite dimension d, and denote by β(V ,G) the minimal number m such that the invariant ring S(V ) is generated by finitely many elements of degree at most m. A classical result of E. Noether says that β(V ,G) 6 |G| provided that charF is coprime to |G|!. If charF divides |G|, then no bounds for β(V ,G) are known except for very special choices of G. In this paper we present a constructive proof of the following. If H 6 G with [G : H] ∈ F∗, and if the restriction V|H is a permutation module (for example, if V is a projective FG-module and H ∈ Sylp(G)), then β(V ,G)6 max{|G|, d(|G| − 1)} regardless of charF.