A new degree bound for vector invariants of symmetric groups

Let R be a commutative ring, V a finitely generated free R-module and G less than or equal to GL(R)(V) a finite group acting naturally on the graded symmetric algebra A = S(V). Let P(V, G) denote the minimal number m, such that the ring A(G) of invariants can he generated by finitely many;elements of degree at most m, For G = Sigma(n) and V(n, k), the k-fold direct sum of the natural permutation module, one knows that beta(V(n,k), Sigma(n)) less than or equal to n, provided that n! is invertible in R. This was used by E. Noether to prove beta(V, G) less than or equal to /G/ if /G/! is an element of R*. In this paper we prove beta(V(n, k), Sigma(n)) less than or equal to max{n, k(n - 1)} for arbitrary commutative rings R and show equality for n = p(s) a prime power and R = Z or any ring with n.1(R) = 0. Our results imply beta(V, G) less than or equal to max{/G/,rank(V)(/G/ - 1)} for any ring with /G/ is an element of R*.