Sharpening the generalized Noether bound in the invariant theory of finite groups

We consider linear representations of a finite group G on a finite dimensional vector space over a field F in which G is invertible. By a theorem due to E. Noether in char 0, and to Fleischmann and Fogarty in general, the ring of invariants is generated by homogeneous elements of degree at most G. Schmid, Domokos, and Heged s sharpened Noetherʹs bound when G is not cyclic and char F=0. We prove that the sharpened bound holds over general fields: If G is not cyclic and G is invertible in F, then the ring of invariants is generated by elements of degree at most if G is even, and at most if G is odd.