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Carter subgroups of finite almost simple groups

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Abstract

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate.

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References

  1. R. W. Carter, “Nilpotent self-normalizing subgroups of soluble groups,” Math. Z., 75, 136–139 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Di Martino and M. C. Tamburini, “I sottogruppi nilpotenti autonormalizzanti di S n e di A n ,” Ist. Lombardo Acc. Sc. Lett. Rend., A, 110, 235–241 (1976).

    Google Scholar 

  3. L. Di Martino and M. C. Tamburini, “Carter subgroups of projective linear groups,” Boll. Un. Mat. Ital., VII. Ser. B, 1, 905–915 (1987).

    MATH  Google Scholar 

  4. N. A. Vavilov, “Nilpotent self-normalizing subgroups of the general linear group over a finite field,” LOMI, 86, 34–39 (1979).

    MathSciNet  Google Scholar 

  5. L. Di Martino, M. C. Tamburini, and A. E. Zalesskii, “Carter subgroups in classical groups,” J. London Math. Soc., II. Ser., 55, No. 2, 264–276 (1997).

    Article  MATH  Google Scholar 

  6. A. Previtali, M. C. Tamburini, and E. P. Vdovin, “The Carter subgroups of some classical groups,” Bull. London Math. Soc., 36, No. 1, 145–155 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. D’aniello, “Sull’ esistenza di sottogruppi nilpotenti autonormalizzanti in alcuni gruppi semplici, II,” Atti Acc. Naz. Lincei, VIII. Ser., Rend. Cl. Sc. Fis. Mat. Nat., 74, 1–6 (1983).

    MATH  MathSciNet  Google Scholar 

  8. F. Dalla Volta, A. Lucchini, and M. C. Tamburini, “On the conjugacy problem for Carter subgroups,” Comm. Alg., 26, No. 2, 395–401 (1998).

    MATH  Google Scholar 

  9. E. P. Vdovin, “On the conjugacy problem for Carter subgroups,” Sib. Mat. Zh., 47, No. 4, 725–730 (2006).

    MathSciNet  Google Scholar 

  10. M. C. Tamburini and E. P. Vdovin, “Carter subgroups of finite groups,” J. Alg., 255, No. 1, 148–163 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Feit and J. Thompson, “Solvability of groups of odd order,” Pac. J. Math., 13, No. 3, 775–1029 (1963).

    MATH  MathSciNet  Google Scholar 

  12. R. W. Carter, Simple Groups of Lie Type, Pure Appl. Math., 28, Wiley, London (1972).

    MATH  Google Scholar 

  13. J. E. Humphreys, Linear Algebraic Groups, Springer, New York (1972).

    Google Scholar 

  14. R. Steinberg, “Automorphisms of finite linear groups,” Can. J. Math., 12, No. 4, 606–615 (1960).

    MATH  MathSciNet  Google Scholar 

  15. R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley, New York (1985).

    MATH  Google Scholar 

  16. R. Carter, “Centralizers of semisimple elements in finite classical groups,” Proc. London Math. Soc., III. Ser., 42, No. 1, 1–41 (1981).

    Article  MATH  Google Scholar 

  17. D. Deriziotis, Conjugacy Classes and Centralizers of Semisimple Elements in Finite Groups of Lie Type, Vorlesungen Fachbereich Math. Univ. Essen, 11 (1984).

  18. D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Part I. Chapter A: “Almost simple K-groups,” Math. Surv. Mon., 40,(3), Am. Math. Soc., Providence, R. I. (1998).

    Google Scholar 

  19. R. Steinberg, Endomorphisms of Algebraic Groups, Mem. Am. Math. Soc., 80, Am. Math. Soc., Providence, R. I. (1968).

    Google Scholar 

  20. A. Borel and J. de Siebental, “Les-sous-groupes fermés de rang maximum des groupes de Lie clos,” Comm. Math. Helv., 23, 200–221 (1949).

    Article  MATH  Google Scholar 

  21. E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” Mat. Sb., 30, No. 2, 349–462 (1952).

    MathSciNet  Google Scholar 

  22. J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Math. Surv. Mon., 43, Am. Math. Soc., Providence, R. I. (1995).

    MATH  Google Scholar 

  23. R. W. Carter, “Centralizers of semisimple elements in finite groups of Lie type,” Proc. London Math. Soc., III. Ser., 37, No. 3, 491–507 (1978).

    Article  MATH  Google Scholar 

  24. B. Hartley and G. Shute, “Monomorphisms and direct limits of finite groups of Lie type,” Q. J. Math., Oxford II. Ser., 35, No. 137, 49–71 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  25. A. S. Kondratiev, “Normalizers of Sylow 2-subgroups in finite simple groups,” Mat. Zametki, 78, No. 3, 368–376 (2005).

    MathSciNet  Google Scholar 

  26. A. S. Kondratiev and V. D. Mazurov, “2-Signalizers of finite simple groups,” Algebra Logika, 42, No. 5, 594–623 (2003).

    MathSciNet  Google Scholar 

  27. A. Borel, R. Carter, C. W. Curtis, et al., Seminar on Algebraic Groups and Related Finite Groups, Springer, Berlin (1970).

    Book  MATH  Google Scholar 

  28. R. W. Carter, “Conjugacy classes in the Weyl group,” Comp. Math., 25, No. 1, 1–59 (1972).

    MATH  Google Scholar 

  29. D. Gorenstein and R. Lyons, “The local structure of finite groups of characteristic 2 type,” Mem. Am. Math. Soc., 42, Am. Math. Soc., Providence, R. I. (1983).

    Google Scholar 

  30. W. Feit and G. Zukerman, “Reality properties of conjugacy classes in spin groups and symplectic groups,” in Algebraists’ Homage: Papers in Ring Theory and Related Topics, New Haven (1981), pp. 239–253.

  31. H. N. Ward, “On Ree’s series of simple groups,” Trans. Am. Math. Soc., 121, No. 1, 62–80 (1966).

    Article  MATH  Google Scholar 

  32. V. M. Levchuk and Ja. N. Nuzhin, “Structure of Ree groups,” Algebra Logika, 24, No. 1, 26–41 (1985).

    MathSciNet  Google Scholar 

  33. J. Conway, R. Curtis, S. Norton, et al., Atlas of Finite Groups, Clarendon, Oxford (1985).

    MATH  Google Scholar 

  34. M. J. Wonenburger, “Transformations which are products of involutions,” J. Math. Mech., 16, No. 1, 32–38 (1966).

    MathSciNet  Google Scholar 

  35. G. M. Seitz, “Unipotent elements, tilting modules, and saturation,” Inv. Math., 141, No. 3, 467–502 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  36. P. H. Tiep and A. E. Zalesski, “Real conjugacy classes in algebraic groups and finite groups of Lie type,” J. Group Theory, 8, No. 3, 291–315 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  37. D. J. Robinson, A Course in the Theory of Groups, Springer, New York (1996).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    E. P. Vdovin

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  1. E. P. Vdovin
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Additional information

Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2.

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Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.

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Vdovin, E.P. Carter subgroups of finite almost simple groups. Algebr Logic 46, 90–119 (2007). https://doi.org/10.1007/s10469-007-0010-6

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  • DOI: https://doi.org/10.1007/s10469-007-0010-6

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