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

Publié le :
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},
     pages = {369--438},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {2},
     year = {2013},
     mrnumber = {3114008},
     zbl = {1366.20001},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_2_369_0/}
}
TY  - JOUR
AU  - Müller, Peter
TI  - Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2013
SP  - 369
EP  - 438
VL  - 12
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2013_5_12_2_369_0/
LA  - en
ID  - ASNSP_2013_5_12_2_369_0
ER  - 
%0 Journal Article
%A Müller, Peter
%T Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2013
%P 369-438
%V 12
%N 2
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2013_5_12_2_369_0/
%G en
%F 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, Série 5, Tome 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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

[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

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

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

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

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

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

[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. | EuDML | MR | Zbl

[61] H. WielandtPrimitive Permutationsgruppen vom Grad 2p, Math. Z. 63 (1956), 478–485. | EuDML | MR | Zbl

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