On Degree Bounds for Invariant Rings of Finite Groups over Finite Fields

Let G be a nite group acting as algebra automorphisms on A := FX 1 ; : : : ; Xn]. If F is a eld of characteristic zero, then, due to classical results of Emmy Noether one knows that the invariant ring A G can be generated in degrees less or equal to jGj. If F an arbitrary commutative ring (e.g. a nite eld), the situation is much less satisfying. In this paper we give an outline on known results (with some new proofs) on degree bounds for arbitrary rings and arbitary nite groups. It turns out that for the nite eld Fq with q = p s , A G can be constructed explicitly from A P , if P is a Sylow-p group of G. We also present some new results on the range of nite elds over which Noether's bound holds for subgroups of classical groups.