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Covering numbers for Chevalley groups

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Abstract

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveC rd=G, wherer=rankG.

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References

  • [ACM] Z. Arad, D. Chillag and G. Moran,Groups with a small covering number, in [AH, Chapter 4].

  • [AFM] Z. Arad, E. Fisman and M. Muzychuk,Order evaluation of products of subsets in finite groups and its applications I, Journal of Algebra182 (1996), 577–603.

    Article  MATH  MathSciNet  Google Scholar 

  • [AH] Z. Arad and M. Herzog,Products of Conjugacy Classes in Groups, Lecture Notes in Mathematics1112, Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  • [AHS] Z. Arad, M. Herzog and J. Stavi,Powers and products of conjugacy classes, in [AH, Chapter 1].

  • [Bo] A. Borel,Linear Algebraic Groups, 2nd enlarged edition, Graduate Texts in Mathematics126, Springer-Verlag, New York, 1991.

    MATH  Google Scholar 

  • [Bou] N. Bourbaki,Groupes et Algèbres de Lie IV, V, VI, Hermann, Paris, 1968.

    Google Scholar 

  • [Car1] R. W. Carter,Simple Groups of Lie Type, Wiley, London, 1989.

    MATH  Google Scholar 

  • [Car2] R. W. Carter,Finite Groups of Lie Type, Wiley, Chichester, 1993.

    Google Scholar 

  • [Dv] Y. Dvir,Covering properties of permutation groups, in [AH, Chapter 3].

  • [EGI] E. W. Ellers and N. L. Gordeev,Gauss decomposition with prescribed semisimple part in classical Chevalley groups, Communications in Algebra22 (1994), 5935–5950.

    Article  MATH  MathSciNet  Google Scholar 

  • [EGII] E. W. Ellers and N. L. Gordeev,Gauss decomposition with prescribed semisimple part in Chevalley groups II: exceptional cases, Communications in Algebra23 (1995), 3085–3098.

    Article  MATH  MathSciNet  Google Scholar 

  • [EGIII] E. W. Ellers and N. L. Gordeev,Gauss decomposition with prescribed semisimple part in classical Chevalley groups III: twisted groups, Communications in Algebra24 (1996), 4447–4475.

    Article  MATH  MathSciNet  Google Scholar 

  • [El] E. W. Ellers,Decomposition of equiaffinities into reflections, Geometriae Dedicata6 (1977), 297–304.

    Article  MATH  MathSciNet  Google Scholar 

  • [GLS] D. Gorenstein, R. Lyons and R. Solomon,The Classification of the Finite Simple Groups, Math Surveys and Monographs 40, Number 1, American Mathematical Society, Providence, R.I., 1994.

    Google Scholar 

  • [Go1] N. L. Gordeev,Products of conjugacy classes in algebraic groups I, Journal of Algebra173 (1995), 715–744.

    Article  MATH  MathSciNet  Google Scholar 

  • [Go2] N. L. Gordeev,Products of conjugacy classes in algebraic groups II, Journal of Algebra173 (1995), 745–779.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ka,r] S. Karni,Covering numbers of groups of small order and sporadic groups, in [AH, Chapter 2].

  • [LL] R. Lawther and M. W. Liebeck,On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class, Journal of Combinatorial Theory, Series A, to appear.

  • [Lev] A. Lev,The covering number of the group PSL n (F), Journal of Algebra182 (1996), 60–84.

    Article  MATH  MathSciNet  Google Scholar 

  • [So] A. R. Sourour,A factorization theorem for matrices, Linear and Multilinear Algebra19 (1986), 141–147.

    Article  MATH  MathSciNet  Google Scholar 

  • [St] R. Steinberg,Lectures on Chevalley Groups, Yale University, 1967.

  • [Va] N. A. Vavilov,The geometry of long root subgroups in Chevalley groups, Vestnik Leningradskogo Universiteta Matematika Mekhanika Atronomiya (1988), vyp 1, 8–11, 116; translated in Vestnik Leningrad University Mathematics21 (1988), 5–10.

  • [Z] I. Zisser,The covering numbers of the sporadic simple groups, Israel Journal of Mathematics67 (1989), 217–224.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 100 St. George Street, M5S 3G3, Toronto, Ontario, Canada

    Erich W. Ellers

  2. Department of Mathematics, Russian State Pedagogical University, Moijka 48, 191-186, St. Petersburg, Russia

    Nikolai Gordeev

  3. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Marcel Herzog

Authors
  1. Erich W. Ellers
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  2. Nikolai Gordeev
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  3. Marcel Herzog
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Corresponding author

Correspondence to Erich W. Ellers.

Additional information

Research supported in part by NSERC Canada Grant A7251.

Research supported in part by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.

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Ellers, E.W., Gordeev, N. & Herzog, M. Covering numbers for Chevalley groups. Isr. J. Math. 111, 339–372 (1999). https://doi.org/10.1007/BF02810691

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  • DOI: https://doi.org/10.1007/BF02810691

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