Invariants of finite groups over fields of characteristic p

Let V denote a finite-dimensional K Vector space and let G denote a finite group of K-linear automorphisms of V. Let V-m denote the direct sum of m copies of V and let G act on the symmetric algebra K[V-m] of V-m by the diagonal action on V-m. A result of Noether implies that, if char K=0, then K[V-m](G) can be generated as a K-algebra by polynomials whose degrees are less than or equal to\G\, no matter how large in is. This paper proves that this result no longer holds when the characteristic of K divides \G\. More precisely, it is proved in this case that there is a positive number alpha, depending only on \G\ and char K, such that every set of K-algebra generators of K[V-m](G) contains a generator whose degree is greater than or equal to alpha m. (C) 1996 Academic Press, Inc.