Invariants of Finite Groups over Fields of Characteristicp

AbstractLetVdenote a finite-dimensionalKvector space and letGdenote a finite group ofK-linear automorphisms ofV. LetVmdenote the direct sum ofmcopies ofVand letGact on the symmetric algebraK[Vm] ofVmby the diagonal action onVm. A result of Noether implies that, if charK=0, thenK[Vm]Gcan be generated as aK-algebra by polynomials whose degrees are ⩽|G|, no matter how largemis. This paper proves that this result no longer holds when the characteristic ofKdivides |G|. More precisely, it is proved in this case that there is a positive numberα, depending only on |G| and charK, such that every set ofK-algebra generators ofK[Vm]Gcontains a generator whose degree is ⩾αm