The Noether bound in invariant theory of finite groups

Let R be a commutative ring, V a finitely generated free R-module and G les s than or equal to GL(R)(V) a finite group acting naturally on the graded s ymmetric algebra A = Sym(V). Let beta (A(G)) denote the minimal number m, s uch that the ring A(G) of invariants can be generated by finitely many elem ents of degree at most m. Furthermore, let H <<vertical bar> G be a normal subgroup such that the index \G : H\ is invertible in R. In this paper we p rove the inequality beta (A(G)) less than or equal to beta (A(H)) . \G : H\. For H = 1 and \G\ invertible in R we obtain Noether's bound beta (A(G)) les s than or equal to \G\, which so far had been shown for arbitrary groups on ly under the assumption that the factorial of the group order, \G\!, is inv ertible in R. (C) 2000 Academic Press.