The Noether Bound in Invariant Theory of Finite Groups

Let R be a commutative ring, V a finitely generated free R-module and G⩽GLR(V) a finite group acting naturally on the graded symmetric algebra A=Sym(V). Let β(AG) denote the minimal number m, such that the ring AG of invariants can be generated by finitely many elements of degree at most m. Furthermore, let H◁G be a normal subgroup such that the index |G : H| is invertible in R. In this paper we prove the inequality β(A G )⩽β(A H )·|G : H|. For H=1 and |G| invertible in R we obtain Noether's bound β(AG)⩽|G|, which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order, |G|!, is invertible in R.