AG - Invariantentheorie PUTTING THE SQUEEZE ON THE NOETHER GAP THE CASE OF THE ALTERNATING GROUPS

Let ρ : G GL(n, IF) be a faithful representation of the finite group G over the field IF. In 1916 E. Noether proved that for IF of characteristic zero the ring of invariants IF[V ]G is generated as an algebra by the invariant polynomials of degree at most G. This result has been generalized to the case where the characteristic of IF is greater than G, or when the characteristic of IF is prime to the order of G and the group G is solvable. In this note we prove that if Noether’s bound fails in the nonmodular case, then it fails for a finite nonabelian simple group. We then show how yet another reworking of Noether’s argument leads to a proof that Noether’s bound holds in the nonmodular case for the alternating groups.