Finite groups with large Noether number are almost cyclic

TL;DR

This paper characterizes finite groups with large Noether number, showing they are nearly cyclic, and establishes bounds on the indispensable degrees of polynomial invariants for various finite simple groups.

Contribution

It extends previous results by providing an asymptotic characterization of groups with bounded ratio of order to Noether number and bounds the Noether number for simple groups of Lie type and sporadic groups.

Findings

Groups with bounded |G|/β(G) have a characteristic cyclic subgroup of bounded index.

For simple groups of Lie type and sporadic groups, β(S) ≤ |S|^{39/40}.

The ratio |G|/β(G) is bounded if and only if G has a characteristic cyclic subgroup of bounded index.

Abstract

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$. Let $\beta(G)$ denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups $G$ with $|G|/\beta(G)$ at most $2$. We prove an asymptotic extension of their result. Namely, $|G|/\beta(G)$ is bounded for a finite group $G$ if and only if $G$ has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If $S$ is a finite simple group of Lie type or a sporadic group then we have $\beta(S) \leq {|S|}^{39/40}$. We ask a number of questions motivated by our results.