On Generators of Modular Invariant Rings of Nite Groups

Let G be a nite group, let V be an F G-module of nite dimension d, and denote by (V; G) the minimal number m such that the invariant ring S(V) G is generated by nitely many elements of degree at most m. A classical result of E. Noether says that (V; G) jGj provided that char F is coprime to jGj!. If char F divides jGj then no bounds for (V; G) are known except for very special choices of G. In this paper we present a constructive proof of the following: If H G with G : H] 2 F and if the restriction V jH is a permutation module (e.g. if V is a projective F G-module and H 2 Syl p (G)) then (V; G) maxfjGj; d(jGj ? 1)g regardless of char F.