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Israel Journal of Mathematics

, Volume 186, Issue 1, pp 125–195 | Cite as

Additive polynomials for finite groups of Lie type

  • Maximilian AlbertEmail author
  • Annette Maier
Article

Abstract

This paper provides a realization of all classical finite groups of Lie type as well as a number of exceptional ones (with low-dimensional representations) as Galois groups over function fields over F q and derives explicit additive polynomials for the extensions. Our unified approach is based on results of Matzat which give bounds for Galois groups of Frobenius modules and uses the structure and representation theory of the corresponding connected linear algebraic groups.

Keywords

Maximal Subgroup Characteristic Polynomial Simple Root Galois Group Maximal Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Abh01]
    S. S. Abhyankar and N. F. J. Inglis, Galois groups of some vectorial polynomials, Transactions of the American Mathematical Society 353 (2001), 2941–2969.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Alb07]
    M. Albert, Classical Groups as Galois Groups in Positive Characteristic, Diplomarbeit, Heidelberg, 2007.Google Scholar
  3. [BCP97]
    W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, Journal of Symbolic Computation 24 (1997), 235–265.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Car85]
    R. W. Carter, Finite Groups of Lie Type, Wiley Series in Pure and Applied Mathematics, Wiley, New York, 1985.zbMATHGoogle Scholar
  5. [Car89]
    R. W. Carter, Simple Groups of Lie Type, Wiley, New York, 1972, 1989.zbMATHGoogle Scholar
  6. [Con08]
    J. Conway, J. McKay and A. Trojan, Galois groups over function fields of positive characteristic, Proceedings of the American Mathematical Society 138 (2010), 1205–1212.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Coo81]
    B.N. Cooperstein, Maximal subgroups of G 2(2n), Journal of Algebra 70 (1981), 23–36.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [DM87]
    D. I. Deriziotis and G. O. Michler, Character table and blocks of finite simple triality groups 3D4(q), Transactions of the American Mathematical Society 303 (1987), 39–70.MathSciNetzbMATHGoogle Scholar
  9. [Elk97]
    N. D. Elkies, Linearized algebra and finite groups of Lie type, I: Linear and symplectic groups, in Applications of Curves over Finite Fields, (AMS-IMS-SIAM Joint Summer Research Conference, July 1997, Washington, Seattle; M. Fried, ed.; Providence: AMS, 1999), Contemporary Mathematics 245 (1997), 77–108.CrossRefGoogle Scholar
  10. [EP05]
    A. J. Engler and A. Prestel, Valued Fields, Springer, Berlin, 2005.zbMATHGoogle Scholar
  11. [Gag73]
    P. C. Gager, Maximal tori in finite groups of Lie type, Ph.D. Thesis, University of Warwick, 1973.Google Scholar
  12. [Gec03]
    M. Geck, An Introduction to Algebraic Geometry and Algebraic Groups, Oxford University Press, 2003.Google Scholar
  13. [Gos98]
    D. Goss, Basic Structures of Function Field Arithmetic, 2nd edition, Springer, Berlin, 1998.zbMATHGoogle Scholar
  14. [KLM01]
    G. Kemper, F. Luebeck and K. Magaard, Matrix generators for the Ree groups 2 G 2(q), Communications in Algebra 29 (2001), 407–415.MathSciNetCrossRefGoogle Scholar
  15. Kle88]
    P. B. Kleidman, The maximal subgroups of the Chevalley groups G 2(q) with q odd, the Ree groups 2 G 2(q), and their automorphism groups, Journal of Algebra 117 (1988), 30–71.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Kle88b]
    P. B. Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups, Journal of Algebra 115 (1988), 182–199.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [MSW94]
    G. Malle, J. Saxl and T. Weigel, Generation of classical groups, Geometriae Dedicata 49 (1994), 85–116.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Mai08]
    A. Maier, Additive polynomials for twisted groups of Lie type, Diplomarbeit, Heidelberg, 2008.Google Scholar
  19. [Mal03]
    G. Malle, Explicit realization of the Dickson groups G 2(q) as Galois groups, Pacific Journal of Mathematics 212 (2003), 157–167.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [Mat03]
    B. H. Matzat, Frobenius modules and Galois groups, in Galois Theory and Modular Forms, Kluwer, Dordrecht, 2003, pp. 233–267.Google Scholar
  21. [Nor94]
    M. V. Nori, Unramified coverings of the affine line in positive characteristic, in Algebraic Geometry and its Applications, Springer, New York, 1994, pp. 209–212.CrossRefGoogle Scholar
  22. [SMC95]
    J. Sándor, D. S. Mitrinovic and B. Crstici, Handbook of Number Theory I, Springer-Verlag, Berlin, 1995.Google Scholar
  23. [Spr98]
    T. A. Springer, Linear Algebraic Groups, 2nd edition, Birkhäuser, Basel, 1998.zbMATHCrossRefGoogle Scholar
  24. [Ste65]
    R. Steinberg, Regular elements of semi-simple algebraic groups, Publications Mathématiques. Institut de Hautes Études Scientifiques 25 (1965), 49–80.MathSciNetCrossRefGoogle Scholar
  25. [Ste68]
    R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society 80 (1968)Google Scholar
  26. [Suz64]
    M. Suzuki, On a class of doubly transitive groups, Annals of Mathematics 75 (1962), 104–145.CrossRefGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.School of Engineering SciencesUniversity of SouthamptonSouthamptonUK
  2. 2.Lehrstuhl A für MathematikRWTH AachenAachenGermany

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