Finite groups with large Noether number are almost cyclic

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.