Finite groups with large Noether number are almost cyclic
Annales de l'Institut Fourier, to appear, 18 p.

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 β(G) denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups G with |G|/β(G) at most 2. We prove an asymptotic extension of their result. Namely, |G|/β(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 β(S)|S| 39/40 . We ask a number of questions motivated by our results.

Noether, Fleischmann et Fogarty ont montré que si le caractéristique du corps sous-jacent ne divise pas l’ordre |G| d’un groupe fini, alors l’anneau de pôlynomes invariants de G est engendré par des pôlynomes de degré au plus égal à |G|. Notons par β(G) le plus haut degré indispensable pour un tel système de générateurs. Cziszter et Domokos ont récemment décrit les groupes finis G tels que |G|/β(G) est au plus égal à 2. Nous démontrons une extension asymptotique de leur résultat, à savoir que |G|/β(G) est borné pour un groupe fini G si et seulement s’il admet un sous-groupe caractéristique cyclique d’indice borné. Durant la démonstration nous trouvons le résultat surprenant suivant : si S est un groupe fini simple de type de Lie ou l’un des groupes sporadiques alors on a β(S)|S| 39/40 . Nous posons égalament quelques questions motivées par nos résultats.

Received : 2017-07-14
Accepted : 2018-06-25
Classification:  13A50,  20D06,  20D08,  20D99
Keywords: polynomial invariants, Noether bound, simple groups of Lie type
     author = {Heged\H us, P\'al and Mar\'oti, Attila and Pyber, L\'aszl\'o},
     title = {Finite groups with large Noether number are almost cyclic},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Finite groups with large Noether number are almost cyclic. Annales de l'Institut Fourier, to appear, 18 p.

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