Finite groups with large Noether number are almost cyclic

Hegedűs, Pál; Maróti, Attila; Pyber, László

doi:10.5802/aif.3280

Finite groups with large Noether number are almost cyclic
[Les groupes finis ayant un grand nombre de Noether sont presque cycliques]

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

Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1739-1756.

Résumé
Abstract

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 $β (S) \leq {| S |}^{39 / 40}$ . 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 $β (S) \leq {| S |}^{39 / 40}$ . We ask a number of questions motivated by our results.

Reçu le : 2017-07-14
Accepté le : 2018-06-25
Publié le : 2019-09-16

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

Affiliations des auteurs :

Hegedűs, Pál ¹ ; Maróti, Attila ² ; Pyber, László ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Heged\'{u}s, 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},
     pages = {1739--1756},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     doi = {10.5802/aif.3280},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3280/}
}

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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3280/

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