Finite groups with large Noether number are almost cyclic
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1739-1756

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
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3280
Classification:  13A50,  20D06,  20D08,  20D99
Keywords: polynomial invariants, Noether bound, simple groups of Lie type
@article{AIF_2019__69_4_1739_0,
     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},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1739-1756},
     doi = {10.5802/aif.3280},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1739_0}
}
Finite groups with large Noether number are almost cyclic. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1739-1756. doi : 10.5802/aif.3280. https://aif.centre-mersenne.org/item/AIF_2019__69_4_1739_0/

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