Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, p. 369-438

We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities for these monodromy groups, and we give applications to Hilbert’s irreducibility theorem.

Published online : 2019-02-21
Classification:  12E25,  20B15,  12E05,  12F12,  14H30
@article{ASNSP_2013_5_12_2_369_0,
     author = {M\"uller, Peter},
     title = {Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {2},
     year = {2013},
     pages = {369-438},
     zbl = {1366.20001},
     mrnumber = {3114008},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_2_369_0}
}
Müller, Peter. Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, pp. 369-438. http://www.numdam.org/item/ASNSP_2013_5_12_2_369_0/

[1] M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), 44–88. | MR 772471 | Zbl 0549.20011

[2] A. Bochert, Über die Zahl verschiedener Werte, die eine Funktion gegebener Buchstaben durch Vertauschung derselben erlangen kann. Math. Ann. 33 (1889), 584–590. | JFM 21.0141.01 | MR 1510562

[3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265. | MR 1484478 | Zbl 0898.68039

[4] P. J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), 1–22. | MR 599634 | Zbl 0463.20003

[5] P. J. Cameron and W. M. Kantor, Antiflag–transitive collineation groups revisited, Incomplete draft, www.maths.qmul.ac.uk/pjc/odds/antiflag.pdf.

[6] P. J. Cameron and W. M. Kantor, 2–transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra 60 (1979), 384–422. | MR 549937 | Zbl 0417.20044

[7] R. W. Carter, Conjugacy classes in the Weyl group, In: “Seminar on Algebraic Groups and Related Finite Groups”, A. Borel, R. Carter, C. W. Curtis et al. (eds.), Vol. 131, Lecture Notes in Mathematics, Springer–Verlag, Berlin, Heidelberg, G1–G22. | MR 269749 | Zbl 0269.20038

[8] R. W. Carter, “Simple Groups of Lie Type”. Wiley and Sons, London, 1972. | MR 407163 | Zbl 0567.20023

[9] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, “Atlas of Finite Groups”, Oxford University Press, Eynsham, 1985. | MR 827219 | Zbl 0568.20001

[10] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, S. M. and K. Wildanger, KANT V4, J. Symb. Comput. 24 (1996), 267–283. | MR 1484479 | Zbl 0886.11070

[11] J. D. Dixon and B. Mortimer, “Permutation Groups”, Springer-Verlag, New York, 1996. | MR 1409812 | Zbl 0951.20001

[12] W. Feit, On symmetric balanced incomplete block designs with doubly transitive automorphism groups, J. Combinatorial Theory Ser. A 14 (1973), 221–247. | MR 327540 | Zbl 0278.05016

[13] W. Feit, Some consequences of the classification of finite simple groups, In: “The Santa Cruz Conference on Finite Groups”, Vol. 37 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, Rhode Island (1980), 175–181. | MR 604576 | Zbl 0454.20014

[14] W. Feit, R. Lyndon and L. L. Scott, A remark about permutations, J. Combinatorial Theory Ser. A 18 (1975), 234–235. | MR 372000 | Zbl 0297.05021

[15] M. Fried, Exposition on an arithmetic–group-theoretic connection via Riemann’s existence theorem, In: “The Santa Cruz Conference on Finite Groups”, Vol. 37 of Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, Rhode Island (1980), 571–602. | MR 604636 | Zbl 0451.14011

[16] M. Fried and P. Dèbes, Rigidity and real residue class fields, Acta Arith. 56 (1990), 291–323. | MR 1096344 | Zbl 0667.14014

[17] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008). URL http://www.gap-system.org.

[18] D. Gorenstein, “Finite Groups”. Harper and Row, New York–Evanston–London, 1968. | MR 231903 | Zbl 0185.05701

[19] R. Guralnick, Monodromy groups of coverings of curves, In: “Galois Groups and Fundamental Groups”, Vol. 41 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge (2003), pages 1–46. | MR 2012212 | Zbl 1071.20001

[20] R. M. Guralnick and J. G. Thompson, Finite groups of genus zero, J. Algebra 131 (1990), 303–341. | MR 1055011 | Zbl 0713.20011

[21] C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geom. Dedicata 2 (1974), 425–460. | MR 335659 | Zbl 0292.20045

[22] D. G. Higman, Finite permutation groups of rank 3. Math. Z. 86 (1964), 145–156. | MR 186724 | Zbl 0122.03205

[23] B. Huppert, “Endliche Gruppen I”, Springer–Verlag, Berlin Heidelberg, 1967. | MR 224703 | Zbl 0412.20002

[24] B. Huppert and N. Blackburn, “Finite Groups III”, Springer–Verlag, Berlin Heidelberg, 1982. | MR 662826 | Zbl 0514.20002

[25] G. A. Jones, Cyclic regular subgroups of primitive permutation groups, J. Group Theory 5 (2002), 403–407. | MR 1931365 | Zbl 1012.20002

[26] G. A. Jones and A. Zvonkin, Orbits of braid groups on cacti. Mosc. Math. J. 2 (2002), 127–160, 200. | MR 1900588 | Zbl 1008.20030

[27] W. M. Kantor, Linear groups containing a Singer cycle. J. Algebra 62 (1980), 232–234. | MR 561126 | Zbl 0429.20004

[28] P. Kleidman, The maximal subgroups of the Chevalley groups G 2 (q) with q odd, of the Ree groups 2 G 2 (q), and of their automorphism groups, J. Algebra 117 (1988), 30–71. | MR 955589 | Zbl 0651.20020

[29] P. Kleidman, The maximal subgroups of the Steinberg triality groups 3 D 4 (q) and of their automorphism groups. J. Algebra 115 (1988), 182–199. | MR 937609 | Zbl 0642.20013

[30] P. Kleidman and M. W. Liebeck, “The Subgroup Structure of the Finite Classical Groups”, Cambridge University Press, Cambridge, 1990. | MR 1057341 | Zbl 0697.20004

[31] P. Kleidman, R. A. Parker and R. A. Wilson, The maximal subgroups of the Fischer group Fi 23 . J. London Math. Soc. 39 (1989), 89–101. | MR 989922 | Zbl 0629.20006

[32] P. Kleidman and R. A. Wilson, The maximal subgroups of Fi 22 , Math. Proc. Cambridge Philos. Soc. 102 (1987), 17–23. | MR 886431 | Zbl 0622.20008

[33] P. Kleidman and R. A. Wilson, The maximal subgroups of J 4 , Proc. London Math. Soc. 56 (1988), 484–510. | MR 931511 | Zbl 0619.20004

[34] E. Landau, “Handbuch der Lehre von der Verteilung der Primzahlen”, Teubner, Leipzig (1909), Second edition by Chelsea, New York, 1953. | JFM 40.0232.08 | MR 68565 | Zbl 0051.28007

[35] V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443. | MR 360852 | Zbl 0325.20008

[36] S. Lang, “Fundamentals of Diophantine Geometry”, Springer–Verlag, New York, 1983. | MR 715605 | Zbl 0528.14013

[37] M. W. Liebeck, C. E. Praeger and J. Saxl, On the O’Nan–Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396. | MR 929529 | Zbl 0647.20005

[38] M. W. Liebeck and J. Saxl, On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proc. London Math. Soc. 55 (1987), 299–330. | MR 896223 | Zbl 0627.20026

[39] S. Linton, The maximal subgroups of the Thompson group, J. London Math. Soc. 39 (1989), 79–88. | MR 989921 | Zbl 0676.20009

[40] K. Magaard, S. Shpectorov and G. Wang, Generating sets of affine groups of low genus (2011), arXiv:1108.4833. | MR 2953829

[41] G. Malle, Fields of definition of some three point ramified field extensions, In: “The Grothendieck theory of dessins d’enfants”, L. Schneps (ed.), London Math. Soc. Lecture Note Ser. 200, Cambridge Univ. Press (1984), 147–168. | MR 1305396 | Zbl 0871.14021

[42] G. Malle, Multi-parameter polynomials with given Galois group. Algorithmic methods in Galois theory, J. Symbolic Comput. 30 (2000), 717–731. | MR 1800034 | Zbl 0967.12005

[43] G. Malle and B. H. Matzat, “Inverse Galois Theory”, Springer Verlag, Berlin, 1999. | MR 1711577 | Zbl 0940.12001

[44] J. McLaughlin, Some subgroups of SL n (𝔽 2 ). Illinois J. Math. 13 (1969), 108–115. | MR 237660 | Zbl 0179.04901

[45] P. Müller, Primitive monodromy groups of polynomials, In: “Recent developments in the inverse Galois problem”, M. Fried (ed.), Contemp. Math., Amer. Math. Soc. 186 (1995), 385–401. | MR 1352284 | Zbl 0840.12001

[46] P. Müller, Reducibility behavior of polynomials with varying coefficients, Israel J. Math. 94 (1996), 59–91. | MR 1394567 | Zbl 0871.12003

[47] P. Müller, Kronecker conjugacy of polynomials, Trans. Amer. Math. Soc. 350 (1998), 1823–1850. | MR 1458331 | Zbl 0894.11006

[48] P. Müller, Finiteness results for Hilbert’s irreducibility theorem, Ann. Inst. Fourier Grenoble 52 (2002), 983–1015. | Numdam | MR 1926669 | Zbl 1014.12002

[49] P. Müller and H. Völklein, On a question of Davenport, J. Number Theory 58 (1996), 46–54. | MR 1387720 | Zbl 0861.11061

[50] M. Neubauer, On primitive monodromy groups of genus zero and one, I, Comm. Algebra 21 (1993), 711–746. | MR 1204753 | Zbl 0792.20001

[51] R. Ree, A theorem on permutations, J. Combinatorial Theory Ser. A 10 (1971), 174–175. | MR 269519 | Zbl 0221.05033

[52] L. Scott, Matrices and cohomology, Anal. Math. 105 (1977), 473–492. | MR 447434 | Zbl 0399.20047

[53] L. Scott, Representations in characteristic p, In: “The Santa Cruz Conference on Finite Groups”, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R. I., 37 (1980), 319–331. | MR 604599 | Zbl 0458.20039

[54] L. L. Scott, On the n,2n problem of Michael Fried, In: “Proceedings of the Conference on Finite Groups”, Academic Press (1975), 471–475. | Zbl 0365.20002

[55] J.-P. Serre, “Topics in Galois Theory”. Jones and Bartlett, Boston, 1992. | MR 1162313 | Zbl 0746.12001

[56] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Pr. Akad. Wiss. 1 (1929), 41–69. (=Ges. Abh., I, 209–266). | JFM 56.0180.05

[57] R. Steinberg, “Endomorphisms of Linear Algebraic Groups”, Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence, R.I., 1968. | MR 230728 | Zbl 0164.02902

[58] P. H. Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093–2167. | MR 1386030 | Zbl 0901.20031

[59] H. Völklein, “Groups as Galois Groups – an Introduction”, Cambridge University Press, New York, 1996. | MR 1405612 | Zbl 0868.12003

[60] A. Wagner, The faithful linear representation of least degree of S n and A n over a field of characteristic 2, Math. Z. 151 (1976), 127–137. | MR 419581 | Zbl 0321.20008

[61] H. Wielandt, Primitive Permutationsgruppen vom Grad 2p, Math. Z. 63 (1956), 478–485. | MR 75200 | Zbl 0071.02303

[62] H. Wielandt, “Finite Permutation Groups”. Academic Press, New York London, 1964. | MR 183775 | Zbl 0138.02501