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Finite simple groups and localization

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Abstract

The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups.

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References

  1. T. Beth, D. Jungnickel and H. Lenz,Design Theory, Cambridge University Press, 1986.

  2. A. K. Bousfield,Constructions of factorization systems in categories, Journal of Pure and Applied Algebra9 (1976/77), 207–220.

    Article  MathSciNet  Google Scholar 

  3. R. W. Carter,Simple Groups of Lie Type, Pure and Applied Mathematics28, John Wiley & Sons, London-New York-Sydney, 1972.

    MATH  Google Scholar 

  4. C. Casacuberta,On structures preserved by idempotent transformations of groups and homotopy types, inCrystallographic Groups and Their Generalizations (Kortrijk, 1999), Contemporary Mathematics262 (2000), 39–68.

    MathSciNet  Google Scholar 

  5. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson,Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

    MATH  Google Scholar 

  6. B. N. Cooperstein,Minimal degree for a permutation representation of a classical group, Israel Journal of Mathematics30 (1978), 213–235.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. D. Dixon and B. Mortimer,Permutation Groups, Graduate Texts in Mathematics163, Springer-Verlag, Berlin-Heidelberg-New York, 1996.

    MATH  Google Scholar 

  8. E. D. Farjoun,Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics1622, Springer-Verlag, Berlin-Heidelberg-New York, 1996.

    Google Scholar 

  9. R. Göbel, J. L. Rodríguez and S. Shelah,Large localizations of finite simple groups, Journal für die reine und angewandte Mathematik, to appear.

  10. R. Göbel and S. Shelah,Constructing simple groups for localizations, Communications in Algebra, to appear.

  11. D. Gorenstein,Finite Simple Groups. An Introduction to their Classification, The University Series in Mathematics, Plenum Press, New York-London, 1982.

    MATH  Google Scholar 

  12. D. Gorenstein, R. Lyons, and R. Solomon,The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A. Almost Simple K-groups, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.

    Google Scholar 

  13. B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967.

    MATH  Google Scholar 

  14. 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 Algebra117 (1988), 30–71.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. B. Kleidman, R. A. Parker, and R. A. Wilson,The maximal subgroups of the Fischer group Fi23, Journal of the London Mathematical Society (2)39 (1989), 89–101.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. B. Kleidman and R. A. Wilson,The maximal subgroups of J 4, Proceedings of the London Mathematical Society (3)56 (1988), 484–510.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. B. Kleidman and R. A. Wilson,J 3 <E 6(4)and M 12 <E 6(5), Journal of the London Mathematical Society (2)42 (1990), 555–561.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Libman,A note on the localization of finite groups, Journal of Pure and Applied Algebra148 (2000), 271–274.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Libman,Cardinality and nilpotency of localizations of groups and G-modules, Israel Journal of Mathematics117 (2000), 221–237.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. W. Liebeck and J. Saxl,On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proceedings of the London Mathematical Society (3)55 (1987), 299–330.

    MATH  MathSciNet  Google Scholar 

  21. S. A. Linton and R. A. Wilson,The maximal subgroups of the Fischer groups Fi24 and Fi′24, Proceedings of the London Mathematical Society (3)63 (1991), 113–164.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. L. Rodríguez, J. Scherer and A. Viruel,Non-simple localizations of finite simple groups, preprint, 2002.

  23. J. J. Rotman,An Introduction to the Theory of Groups, Fourth edition, Graduate Texts in Mathematics 148, Springer-Verlag, Berlin-Heidelberg-New York, 1995.

    MATH  Google Scholar 

  24. J. Saxl, D. B. Wales and R. A. Wilson,Embedings of Sz(32)in E 8(5), The Bulletin of the London Mathematical Society32 (2000), 196–202.

    Article  MATH  MathSciNet  Google Scholar 

  25. R. A. Wilson,The complex Leech lattice and maximal subgroups of the Suzuki group, Journal of Algebra84 (1983), 151–188.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. A. Wilson,The maximal subgroups of the O’Nan group, Journal of Algebra97 (1985), 467–473.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. A. Wilson,The maximal subgroups of the Baby Monster, I, Journal of Algebra211 (1999), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, E-04120, Almería, Spain

    José L. Rodríguez

  2. Institut de Mathématiques, Université de Lausanne, CH-1015, Lausanne, Switzerland

    Jérôme Scherer & Jacques Thévenaz

Authors
  1. José L. Rodríguez
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  2. Jérôme Scherer
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  3. Jacques Thévenaz
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Corresponding author

Correspondence to José L. Rodríguez.

Additional information

Partially supported by DGESIC grant PB97-0202 and the Swiss National Science Foundation.

Partially supported by the Swiss National Science Foundation.

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Rodríguez, J.L., Scherer, J. & Thévenaz, J. Finite simple groups and localization. Isr. J. Math. 131, 185–202 (2002). https://doi.org/10.1007/BF02785857

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  • DOI: https://doi.org/10.1007/BF02785857

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