Mathematische Annalen

, Volume 315, Issue 3, pp 503–510 | Cite as

Putting the squeeze on the Noether gap – The case of the alternating groups \(A_n\)

  • Larry Smith
Original article


Let \(\rho : G \hookrightarrow{\rm GL}(n, \mathbb{F})\) be a faithful representation of the finite group G over the field \(\mathbb{F}\). In 1916 E. Noether proved that for \(\mathbb{F}\) of characteristic zero the ring of invariants \(\mathbb{F}[V]^G\) is generated as an algebra by the invariant polynomials of degree at most \(|G|\). This upper bound for the degrees of the generators in a minimal generating set is referred to as Noether's bound. The result has been generalized to the case where the characteristic of \(\mathbb{F}\) is greater than the order of G, or when the characteristic of \(\mathbb{F}\) is prime to the order of G, and the group G is solvable. In this note we prove that if Noether's bound fails when the order of the group is prime to the characteristic of \(\mathbb{F}\), referred to as 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.

Mathematics Subject Classification (1991):13A50 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Larry Smith
    • 1
  1. 1.School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA US

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