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Normal subgroups ofSL 1,D and the classification of finite simple groups

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Abstract

LetD be a division algebra of degree three over an algebraic number fieldK and let G = SLD. We prove that the normal subgroup structure of G(K) is given by the Platonov-Margulis conjecture. The proof uses the classification of finite simple groups.

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References

  1. Conway J, Curtis R, Norton S, Parker R and Wilson R,Atlas of Finite Groups (Oxford: Clarendon Press) (1985)

    MATH  Google Scholar 

  2. Bourbaki N,Groupes et Algèbres de Lie (Hermann, Paris) (1968)

    MATH  Google Scholar 

  3. Bergelson V and Shapiro D, Multiplicative subgroups of finite index in a ring,Proc. AMS 116 (1992) 885–896

    Article  MATH  MathSciNet  Google Scholar 

  4. Feit W and Thompson J G, Solvability of groups of odd order,Pac. J. Math. 13 (1963) 755–1029

    MathSciNet  Google Scholar 

  5. Hall M,The Theory of Groups (New York: The Macmillan Company) (1959)

    MATH  Google Scholar 

  6. Kneser M, Orthogonale Gruppen über algebraicshen Zahlköpern,J. Reine Angew. Math. 196 (1956) 213–220

    MATH  MathSciNet  Google Scholar 

  7. Margulis G A, Finiteness of quotient groups of discrete subgroups,Funct. Anal., i ego Prilozhenia 13 (1979) 28–39. English translation:Fund. Anal. Appl. 13 (1979) 178–187

    MATH  MathSciNet  Google Scholar 

  8. Margulis G A, On the multiplicative group of a quaternion algebra over a global field.Dokl. Akad. Nauk SSSR 252 (1980) 542–546. English translation:Sov. Math. Dokl. 21 (1980) 780–784

    MathSciNet  Google Scholar 

  9. Platonov V P and Rapinchuk A S, On the group of rational points of three-dimensional groups.Dokl. Akad. Sci. USSR 247 (1979) 279–282. English translation:Sov. Math. Dokl. 20 (1979) 693–697

    MathSciNet  Google Scholar 

  10. Platonov V P and Rapinchuk A S, The multiplicative structure of division algebras over number fields and the Hasse norm principle.Trudy Math. In-ta Acad. Sci. USSR 165 (1984) 171–187. English translation:Proc. Steklov Inst. Math. 165 (1985) 187–205

    MATH  MathSciNet  Google Scholar 

  11. Platonov V P and Rapinchuk A S,Algebraic Groups and Number Theory (Moscow: Nauka Publishers) (1991) (English translation:Pure Appl. Math. Ser. N139, Academic Press) (1993)

    MATH  Google Scholar 

  12. Raghunathan M S, On the group of norm 1 elements in a division algebra,Math. Ann. 279 (1988) 457–484

    Article  MATH  MathSciNet  Google Scholar 

  13. Rapinchuk A S, Benyash-Krivetz V V, Chernousov V I, Representation varieties of the fundamental groups of compact orientable surfaces, Preprint 94-030, SFB 343, Universität Bielefeld (1994)

  14. Riehm C, The norm 1 group of p-adic division algebra,Am. J. Math. 92 (1970) 499–523

    Article  MATH  MathSciNet  Google Scholar 

  15. Springer T A and Steinberg R, Conjugacy classes In: Seminar on Algebraic Groups and Related Finite Groups,Lect. Notes in Math. 131 (1970) 167–266 (Berlin, Heidelberg, NY Springer)

    Google Scholar 

  16. Steinberg R, Lectures on Chevalley groups,Yale Univ. Lecture Notes (1967)

  17. Tomanov G, On the reduced norm 1 group of a division algebra over a global field,Izv. Acad. Sci. USSR, Ser. Math. 55 (1991). English translation:Math. USSR Izvestiya 39 (1992) 895–904

  18. Turnwald G, Multiplicative subgroups of finite index in a division ring,Proc. AMS 120 (1994) 377–381

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, 22903, Charlottesville, Virginia, USA

    Andrei Rapinchuk & Alexander Potapchik

  2. SFB 343, Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    Alexander Potapchik

  3. Department of Mathematics, University of Toronto, M5S1A1, Toronto, Ontario, Canada

    Alexander Potapchik

Authors
  1. Andrei Rapinchuk
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  2. Alexander Potapchik
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Rapinchuk, A., Potapchik, A. Normal subgroups ofSL 1,D and the classification of finite simple groups. Proc. Indian Acad. Sci. (Math. Sci.) 106, 329–368 (1996). https://doi.org/10.1007/BF02837693

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  • DOI: https://doi.org/10.1007/BF02837693

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