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Surjective word maps and Burnside’s \(p^aq^b\) theorem

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Abstract

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map \((x,y) \mapsto x^Ny^N\) is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map \((x,y,z) \mapsto x^Ny^Nz^N\) is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map \((x,y) \mapsto x^Ny^N\) that depend on the number of prime factors of the integer N.

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References

  1. Alladi, K., Solomon, R.M., Turull, A.: Finite simple groups of bounded subgroup chain length. J. Algebra 231, 374–386 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertram, E.: Even permutations as a product of two conjugate cycles. J. Comb. Theory Ser. A 12, 368–380 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brundan, J., Kleshchev, A.S.: Lower bounds for degrees of irreducible Brauer characters of finite general linear groups. J. Algebra 223, 615–629 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burnside, W.: On groups of order \(p^{\alpha }q^{\beta }\). Proc. Lond. Math. Soc. S2–1, 388–392 (1904)

    Article  MATH  Google Scholar 

  6. Carter, R.W.: Conjugacy classes in the Weyl group. In: Seminar on Algebraic Groups and Related Finite Groups, Springer Lecture Notes 131. pp. 297–318 (1970)

  7. Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, Chichester (1985)

    MATH  Google Scholar 

  8. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

    MATH  Google Scholar 

  9. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103, 103–161 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Digne, F., Michel, J.: Representations of finite groups of lie type. London Mathematical Society Student Texts, vol. 21. Cambridge University Press, Cambridge (1991)

  11. Ellers, E.W., Gordeev, N.: On the conjectures of J. Thompson and O. Ore. Trans. Am. Math. Soc. 350, 3657–3671 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Enomoto, H.: The characters of the finite symplectic groups \(Sp(4, q)\), \(q = 2^{f}\). Osaka J. Math. 9, 75–94 (1972)

    MathSciNet  MATH  Google Scholar 

  13. Feit, W., Thompson, J.G.: Solvability of groups of odd order. Pac. J. Math. 13, 775–1029 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geck, M., Hiss, G., Lübeck, F., Malle, G., Pfeiffer, G.: CHEVIE-a system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras. Appl. Algebra Eng. Commun. Comput. 7, 175–210 (1996)

    Article  MATH  Google Scholar 

  15. Gluck, D.: Sharper character value estimates for groups of Lie type. J. Algebra 174, 229–266 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gow, R.: Commutators in finite simple groups of Lie type. Bull. Lond. Math. Soc. 32, 311–315 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guralnick, R.M.: Low-dimensional representations of special linear groups in cross characteristics. Proc. Lond. Math. Soc. 78, 116–138 (1999)

    Article  MathSciNet  Google Scholar 

  18. Guralnick, R.M., Tiep, P.H.: Cross characteristic representations of even characteristic symplectic groups. Trans. Am. Math. Soc. 356, 4969–5023 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guralnick, R.M., Tiep, P.H.: Effective results on the Waring problem for finite simple groups. Am. J. Math. 137, 1401–1430 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guralnick, R.M., Tiep, P.H.: Lifting in Frattini covers and a characterization of finite solvable groups. J. Reine Angew. Math. 708, 49–72 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Guralnick, R.M., Malle, G.: Products of conjugacy classes and fixed point spaces. J. Am. Math. Soc. 25, 77–121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guralnick, R.M., Lübeck, F.: On \(p\)-singular elements in Chevalley groups in characteristic \(p\). In: Groups and Computation, III. (Columbus, OH, 1999), pp. 169–182, Ohio State University Mathematical Research Institute Publications, vol. 8, de Gruyter, Berlin (2001)

  23. Guralnick, R.M., Larsen, M., Tiep, P.H.: Representation growth in positive characteristic and conjugacy classes of maximal subgroups. Duke Math. J. 161, 107–137 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guralnick, R.M., Malle, G., Tiep, P.H.: Product of conjugacy classes in finite simple classical groups. Adv. Math. 234, 618–652 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kleidman, P.B., Liebeck, M.W.: The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge (1990)

  26. Kleshchev, A.S., Sin, P., Tiep, P.H.: Representations of the alternating group which are irreducible over subgroups. II. Am. J. Math. 138, 1383–1423 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Landazuri, V., Seitz, G.: On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32, 418–443 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Larsen, M., Shalev, A.: Characters of symmetric groups: sharp bounds and applications. Invent. Math. 174, 645–687 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Larsen, M., Shalev, A.: Word maps and Waring type problems. J. Am. Math. Soc. 22, 437–466 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Larsen, M., Shalev, A., Tiep, P.H.: The Waring problem for finite simple groups. Ann. Math. 174, 1885–1950 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Larsen, M., Shalev, A., Tiep, P.H.: Waring problem for finite quasisimple groups. Int. Math. Res. Not. 10, 2323–2348 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, H.: Triples in Finite Groups and a Conjecture of Guralnick and Tiep. Ph. D. Thesis, University of Arizona (2017)

  33. Liebeck, M.W., Seitz, G.M.: Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras. Mathematical Surveys and Monographs, vol. 180. American Mathematical Society, Providence, RI (2012)

  34. Liebeck, M.W., Shalev, A.: Diameters of finite simple groups: sharp bounds and applications. Ann. Math. 154, 383–406 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liebeck, M.W., Shalev, A.: Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks. J. Algebra 276, 552–601 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liebeck, M.W., Saxl, J., Seitz, G.M.: Subgroups of maximal rank in finite exceptional groups of Lie type. Proc. Lond. Math. Soc. 65, 297–325 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liebeck, M.W., O’Brien, E.A., Shalev, A., Tiep, P.H.: The Ore conjecture. J. Eur. Math. Soc. 12, 939–1008 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liebeck, M.W., O’Brien, E.A., Shalev, A., Tiep, P.H.: Products of squares in finite simple groups. Proc. Am. Math. Soc. 140, 21–33 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lübeck, F.: Character degrees and their multiplicities for some groups of Lie type of rank \(<9\). http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html

  40. Lübeck, F.: Numbers of conjugacy classes in some series of finite groups of Lie type. http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/nrclasses/nrcldata.html

  41. Lübeck, F.: Smallest degrees of representations of exceptional groups of Lie type. Commun. Algebra 29, 2147–2169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lübeck, F., Malle, G.: \((2,3)\)-generation of exceptional groups. J. Lond. Math. Soc. 59, 109–122 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lusztig, G.: Unipotent characters of the symplectic and odd orthogonal groups over a finite field. Invent. Math. 64, 263–296 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  44. Magaard, K., Tiep, P.H.: Irreducible tensor products of representations of quasi-simple finite groups of Lie type. In: Collins, M.J., Parshall, B.J., Scott, L.L. (eds.) Modular Representation Theory of Finite Groups, pp. 239–262. Walter de Gruyter, Berlin (2001)

    Google Scholar 

  45. Malcolm, A.: The involution width of finite simple groups. J. Algebra 493, 297–340 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Malle, G., Saxl, J., Weigel, T.: Generation of classical groups. Geom. Dedicata 49, 85–116 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. Navarro, G., Tiep, P.H.: Rational irreducible characters and rational conjugacy classes in finite groups. Trans. Am. Math. Soc. 360, 2443–2465 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nguyen, H.N.: Low-dimensional complex characters of the symplectic and orthogonal groups. Commun. Algebra 38, 1157–1197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Nozawa, S.: On the characters of the finite general unitary group \(U(4, q^{2})\). J. Fac. Sci. Univ. Tokyo Sect. IA 19, 257–295 (1972)

    MathSciNet  MATH  Google Scholar 

  50. Nozawa, S.: Characters of the finite general unitary group \(U(5, q^{2})\). J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 23, 23–74 (1976)

    MathSciNet  MATH  Google Scholar 

  51. Rosser, B.: Explicit bounds for some functions of prime numbers. Am. J. Math. 63, 211–232 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  52. Segal, D.: Words: Notes on Verbal Width in Groups. London Mathematical Society Lecture Note Series, vol. 361. Cambridge University Press, Cambridge (2009)

  53. Shalev, A.: Word maps, conjugacy classes, and a noncommutative Waring-type theorem. Ann. Math. 170, 1383–1416 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  54. Sin, P., Tiep, P.H.: Rank \(3\) permutation modules for finite classical groups. J. Algebra 291, 551–606 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Spaltenstein, N.: Caractères unipotents de \({}^{3} D_4(F_q)\). Comment. Math. Helv. 57, 676–691 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  56. Springer, T.A., Steinberg, R.: Conjugacy classes, In: Seminar on Algebraic Groups and Related Finite Groups, Springer Lecture Notes 131. pp 168–266 (1970)

  57. Srinivasan, B.: The characters of the finite symplectic group \(Sp(4, q)\). Trans. Am. Math. Soc. 131, 488–525 (1968)

    MathSciNet  MATH  Google Scholar 

  58. Steinberg, R.: The representations of \(GL(3, q)\), \(GL(4, q)\), \(PGL(3, q)\), and \(PGL(4, q)\). Can. J. Math. 3, 225–235 (1951)

    Article  MATH  Google Scholar 

  59. The GAP group, GAP—groups, algorithms, and programming, Version 4.8.7, 2017. http://www.gap-system.org

  60. Tiep, P.H.: Dual pairs and low-dimensional representations of finite classical groups. Preprint

  61. Tiep, P.H., Zalesskii, A.: Minimal characters of the finite classical groups. Commun. Algebra 24, 2093–2167 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tiep, P.H., Zalesskii, A.E.: Some characterizations of the Weil representations of the symplectic and unitary groups. J. Algebra 192, 130–165 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  63. Tiep, P.H., Zalesskii, A.E.: Real conjugacy classes in algebraic groups and finite groups of Lie type. J. Group Theory 8, 291–315 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  64. Unger, W.R.: Computing the character table of a finite group. J. Symb. Comput. 41, 847–862 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ward, H.N.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)

    MathSciNet  MATH  Google Scholar 

  66. Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author was partially supported by the NSF Grants DMS-1302886 and DMS-1600056, and the Simons Foundation Fellowship 224965. He also thanks the Institute for Advanced Study for its support. The third author was partially supported by the Marsden Fund of New Zealand via Grant UOA 1626. The fourth author was supported by ERC Advanced Grant 247034, ISF Grant 1117/13 and the Vinik Chair of Mathematics which he holds. The fifth author was partially supported by the NSF Grants DMS-1201374 and DMS-1665014, the Simons Foundation Fellowship 305247, the Mathematisches Forschungsinstitut Oberwolfach, and the EPSRC. Parts of the paper were written while the fifth author visited the Department of Mathematics, Harvard University, and Imperial College, London. He thanks Harvard University and Imperial College for generous hospitality and stimulating environments.

      The authors thank Frank Lübeck for providing them with the proof of Lemma 7.8.

      The authors are grateful to the referee for careful reading and insightful comments on the paper that helped improve both the exposition of the paper and some of its results.

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Guralnick, R.M., Liebeck, M.W., O’Brien, E.A. et al. Surjective word maps and Burnside’s \(p^aq^b\) theorem. Invent. math. 213, 589–695 (2018). https://doi.org/10.1007/s00222-018-0795-z

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