Finite groups with large Noether number are almost cyclic
[Les groupes finis ayant un grand nombre de Noether sont presque cycliques]
Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1739-1756.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3280
Classification : 13A50, 20D06, 20D08, 20D99
Keywords: polynomial invariants, Noether bound, simple groups of Lie type
Mot clés : polynomes invariants, majorant de Noether, groupes simples de type de Lie
Hegedűs, Pál 1 ; Maróti, Attila 2 ; Pyber, László 2

1 Department of Mathematics Central European University Nádor utca 9 H-1051 Budapest, Hungary
2 Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 H-1053, Budapest, Hungary
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Hegedűs, Pál; Maróti, Attila; Pyber, László. Finite groups with large Noether number are almost cyclic. Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1739-1756. doi : 10.5802/aif.3280. http://www.numdam.org/articles/10.5802/aif.3280/

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