Groups with large Noether bound
[Les groupes pour lesquels la borne de Noether est grande]
Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 909-944.

Nous classifions les groupes finis ayant un invariant polynômial indécomposable de degré au moins la moitié de l’ordre du groupe. Il est démontré qu’en exceptant quatre groupes particuliers, ce sont exactement les groupes avec un sous-groupe cyclique d’indice au plus deux.

The finite groups having an indecomposable polynomial invariant of degree at least half the order of the group are classified. It turns out that –apart from four sporadic exceptions– these are exactly the groups with a cyclic subgroup of index at most two.

DOI : 10.5802/aif.2868
Classification : 13A50, 11B50
Keywords: Noether bound, polynomial invariant, zero-sum sequence
Mot clés : La borne de Noether, invariants polynômiaux, suites de somme nulle
Cziszter, Kálmán 1 ; Domokos, Mátyás 2

1 Central European University, Department of Mathematics and its Applications, Nádor u. 9, 1051 Budapest, Hungary
2 Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053 Budapest, Hungary
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Cziszter, Kálmán; Domokos, Mátyás. Groups with large Noether bound. Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 909-944. doi : 10.5802/aif.2868. https://aif.centre-mersenne.org/articles/10.5802/aif.2868/

[1] Benson, D. J. Polynomial Invariants of Finite Groups, Cambride University Press, 1993 | MR | Zbl

[2] Berkovich, Y. Groups of Prime Power Order, de Gruyter Expositions in Mathematics, I, de Gruyter, Berlin, New York, 2008 | Zbl

[3] Brown, K. Cohomology of Groups, GTM, 87, Springer, 1982 | MR | Zbl

[4] Bryant, R. M.; Kemper, G. Global degree bounds and the transfer principle, J. Algebra, Volume 284 (2005) no. 1, pp. 80-90 | MR | Zbl

[5] Burnside, W. Theory of Groups of Finite Order, Cambridge University Press, 1911

[6] Collins, M. J. The characterization of the Suzuki groups by their Sylow 2-subgroups, Math. Z., Volume 123 (1971), pp. 32-48 | MR | Zbl

[7] Cziszter, K. The Noether number of the non-abelian group of order 3p, Periodica Math. Hung., Volume 68 (2014), pp. 150-159 | MR

[8] Cziszter, K.; Domokos, M. On the generalized Davenport constant and the Noether number, Cent. Eur. J. Math., Volume 11 (2013) no. 9, pp. 1605-1615 | DOI | MR | Zbl

[9] Cziszter, K.; Domokos, M. The Noether bound for the groups with a cyclic subgroup of index two, J. Algebra, Volume 399 (2014), pp. 546-560 | MR

[10] Delorme, Ch.; Ordaz, O.; Quiroz, D. Some remarks on Davenport constant, Discrete Mathematics, Volume 237 (2001), pp. 119-128 | MR | Zbl

[11] Derksen, H.; Kemper, G. Computational Invariant Theory, Encyclopedia of Mathematical Sciences, 130, Springer-Verlag, 2002 | MR | Zbl

[12] Derksen, H.; Kemper, G. On Global Degree Bounds for Invariants, CRM Proceedings and Lecture Notes, Volume 35 (2003), pp. 37-41 | MR | Zbl

[13] Dixmier, Jacques Sur les invariants du groupe symétrique dans certaines représentations. II, Topics in invariant theory (Paris, 1989/1990) (Lecture Notes in Math.), Volume 1478, Springer, Berlin, 1991, pp. 1-34 | DOI | MR | Zbl

[14] Domokos, M.; Hegedűs, P. Noether’s bound for polynomial invariants of finite groups, Arch. Math. (Basel), Volume 74 (2000) no. 3, pp. 161-167 | MR | Zbl

[15] Fleischmann, P. On invariant theory of finite groups, Invariant theory in all characteristics (CRM Proc. Lecture Notes), Volume 35, Amer. Math. Soc., Providence, RI, 2004, pp. 43-69 | MR | Zbl

[16] Fleishmann, P. The Noether bound in invariant theory of finite groups, Adv. Math., Volume 156 (2000) no. 1, pp. 23-32 | MR | Zbl

[17] Fogarty, J. On Noether’s bound for polynomial invariants of a finite group, Electron. Res. Announc. Amer. Math. Soc., Volume 7 (2001), pp. 5-7 | MR | Zbl

[18] Gao, W.; Geroldinger, A. Zero-sum problems in finite abelian groups: a survey, Expo. Math., Volume 24 (2006), pp. 337-369 | MR | Zbl

[19] Geroldinger, A.; Halter-Koch, F. Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Monographs and textbooks in pure and applied mathematics, Chapman & Hall/CRC, 2006 | MR | Zbl

[20] Göbel, M. Computing bases of permutation-invariant polynomials, J. Symbolic Computation, Volume 19 (1995), pp. 285-291 | MR | Zbl

[21] Grosshans, F.D. Vector invariants in arbitrary characteristic, Transformation Groups, Volume 12 (2007), pp. 499-514 | MR | Zbl

[22] Halter-Koch, F. A generalization of Davenport’s constant and its arithmetical applications, Colloquium Mathematicum, Volume LXIII (1992), pp. 203-210 | MR | Zbl

[23] Higman, G. Suzuki 2-groups, Illinois Journal of Mathematics, Volume 7 (1963), pp. 79-95 | MR | Zbl

[24] Hilbert, D. Über die Theorie der algebraischen Formen, Math. Ann., Volume 36 (1890), pp. 473-531 | MR

[25] Huffman, W. C. Polynomial Invariants of Finite linear Groups of degree two, Canad. J. Math, Volume 32 (1980), pp. 317-330 | MR | Zbl

[26] Huppert, B. Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967 | MR | Zbl

[27] Kemper, G. Separating invariants, Journal of Symbolic Computation, Volume 44 (2009) no. 9, pp. 1212-1222 | MR | Zbl

[28] Knop, F. On Noether’s and Weyl’s bound in positive characteristic, Invariant theory in all characteristics (CRM Proc. Lecture Notes), Volume 35, Amer. Math. Soc., Providence, RI, 2004, pp. 175-188 | MR | Zbl

[29] Neusel, M.; Smith, L. Invariant Theory of Finite Groups, AMS, 2001 | MR | Zbl

[30] Noether, E. Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., Volume 77 (1916), pp. 89-92 | MR

[31] Noether, E. Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p, Nachr. Ges. Wiss. Göttingen (1926), pp. 28-36

[32] Pawale, V. M. Invariants of semi-direct products of cyclic groups (1999) (Ph.D. Thesis, Brandeis University)

[33] Popov, V. L.; Vinberg, E.B. Invariant Theory, Algebraic Geometry IV (Encyclopedia of Mathematical Sciences), Volume 55, Springer-Verlag, Berlin-Heidelberg, 1994 | Zbl

[34] Richman, D. R. Invariants of finite groups over fields of characteristic p, Adv. Math., Volume 124 (1996), pp. 25-48 | MR | Zbl

[35] Roquette, P. Realisierung von Darstellungen endlicher nilpotenten Gruppen, Arch. Math., Volume 9 (1958), pp. 241-250 | MR | Zbl

[36] Schmid, B. J. Finite groups and invariant theory, Topics in invariant theory (Paris, 1989/1990) (Lecture Notes in Math.), Volume 1478, Springer, Berlin, 1991, pp. 35-66 | DOI | MR | Zbl

[37] Serre, J. P. Representations linéares des groupes finis, Hermann, Paris, 1998 | Zbl

[38] Sezer, M. Sharpening the generalized Noether bound in the invariant theory of finite groups, J. Algebra, Volume 254 (2002) no. 2, pp. 252-263 | MR | Zbl

[39] Dias da Silva, J. A.; Hamidoune, Y. O. Cyclic Spaces for Grassmann Derivatives and Additive Theory, Bull. London Math. Soc., Volume 26 (1994) no. 2, pp. 140-146 | MR | Zbl

[40] Thompson, J. G. Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 383-437 | MR | Zbl

[41] Wehlau, D. The Noether number in invariant theory, Comptes Rendus Math. Rep. Acad. Sci. Canada, Volume 28 (2006) no. 2, pp. 39 - 62 | MR | Zbl

[42] Weyl, H. The Classical Groups, Princeton University Press, Princeton, 1939 | MR

[43] Zassenhaus, H. Über endliche Fastkörper, Abhandlungen aus dem Mathematischen Seminar der Hamburgische Universität, Volume 11 (1935), pp. 187-220 | MR | Zbl

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