Skip to main content

Simple Locally Finite Groups of Finite Morley Rank and Odd Type

  • Chapter
Finite and Locally Finite Groups

Part of the book series: NATO ASI Series ((ASIC,volume 471))

Abstract

The paper is devoted to a discussion of possible approaches to the classification of simple infinite groups of finite Morley rank. As an application of methods borrowed from finite group theory we classify locally finite simple groups of finite Morley rank with Černikov Sylow 2-subgroups (without using the classification of finite simple groups).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Altinel, Groups of Finite Morley Rank with Strongly Embedded Subgroups, Ph. D. thesis, Rutgers University, 1994.

    Google Scholar 

  2. T. Altinel, A. Borovik and G. Cherlin, Groups of mixed type, in preparation.

    Google Scholar 

  3. T. Altmel, G. Cherlin, L.-J. Corredor and A. Nesin, A Hall theorem for ω-stable groups, to appear.

    Google Scholar 

  4. A. O. Asar, On a problem of Kegel and Wehrfritz, J. Algebra. 59 (1979), 47–55.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Aschbacher, Finite groups with a proper 2-generated core, Trans. Aimer. Matk. Soc. 197 (1974), 87–112.

    MathSciNet  MATH  Google Scholar 

  6. M. Aschbacher, On finite groups of component type, Illinois J. Math. 19 (1975), 87–115.

    MathSciNet  MATH  Google Scholar 

  7. M. Aschbacher, A characterization of Chevalley groups over fields of odd order. I, II, Ann. Math. 106 (1977), 353–398, 399–468. Corrections: Ann. Math. 111 (1980), 411–414.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Aschbacher and G. M. Seitz, On groups with a standard component of known type, Osaka J. Math. 13 (1976), 439–482.

    MathSciNet  MATH  Google Scholar 

  9. J. T. Baldwin and J. Saxl, Logical stability in group theory, J. Austral. Math. Soc. (Ser. A) 21 (1976), 267–276.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Baudisch, A new uncountably categorical pure group, I, II, preprints A93–40 and A94-23, Freie Universität Berlin, 1993, 1994.

    Google Scholar 

  11. O. V. Belegradek, On groups of finite Morley rank, in Abstracts of the Eighth International Congress of Logic, Methodology and Philosophy of Science, LMPS ’87, Moscow, USSR, 17–22 August 1987, Moscow, 1987, pp. 100–102.

    Google Scholar 

  12. V. V. Belyaev, On locally finite Chevalley groups, in 17–th All-Union Algebraic Conference, Leningrad, 1981, Part 2, p. 17 (in Russian).

    Google Scholar 

  13. V. V. Belyaev, Locally finite Chevalley groups, in Investigations in Group Theory, Urals Scientific Centre, Sverdlovsk, 1984, pp. 39–50 (in Russian).

    Google Scholar 

  14. H. Bender, On groups with abelian Sylow 2-subgroups, Math. Z. 117 (1970), 164–176.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Bender, Transitive Gruppe gerader ordnung, in denen jede Involution genau einen Punkt festlasst, J. Algebra 17 (1971), 175–204.

    Article  MathSciNet  Google Scholar 

  16. H. Bender, Goldschmidt’s 2-signalizer functor theorem, Israel J. Math. 22 (1975), 208–213.

    Article  MathSciNet  Google Scholar 

  17. A. Borel and J. Tits, Groupes réductifs, Publ. IHES 27 (1965), 55–151.

    MathSciNet  Google Scholar 

  18. A. V. Borovik, Periodic linear groups of odd characteristic, Soviet Math. Dokl. 26 (1982), 484–486.

    MATH  Google Scholar 

  19. A. V. Borovik, Involutions in groups with dimension, preprint (Acad. Nauk SSSR, Sibirsk. Otdel. Vychisl. Tsentr), No. 512, Novosibirsk, 1982 (in Russian).

    Google Scholar 

  20. A. V. Borovik, Bmbeddings of finite Chevalley groups and periodic linear groups, Siberian Math. J. 24 (1983), 843–851.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. V. Borovik, Classification of periodic linear groups over fields of odd characteristic, Siberian Math. J. 25 (1984), 217–235.

    Article  MathSciNet  Google Scholar 

  22. A. V. Borovik, On signalizer functors for groups of finite Morley rank, in Soviet-French Colloquium on Model Theory, Karaganda, 1990, p. 11 (in Russian).

    Google Scholar 

  23. A. Borovik and A. Nesin, On the Schur-Zassenhaus theorem for groups of finite Morley rank, J. Symbolic Logic 57 (1992), 1469–1477.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. V. Borovik and A. Nesin, Groups of Finite Morley Rank, Oxford University Press, 1994.

    MATH  Google Scholar 

  25. A. Borovik and A. Nesin, Schur-Zassenhaus theorem revisited, J. Symbolic Logic 59 (1994), 283–391.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. V. Borovik and B. Poizat, Tores et p-groupes, J. Symbolic Logic 55 (1990), 478–491.

    Article  MathSciNet  MATH  Google Scholar 

  27. N. Burgoyne and C. Williamson, Centralizers of semisimple groups, Pacific. J. Math. 72 (1977), 341–350.

    MathSciNet  MATH  Google Scholar 

  28. R. Carter, Simple Groups of Lie Type, Wiley-Interscience, 1972 .

    MATH  Google Scholar 

  29. R. Carter, Finite groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley-Interscience,1985.

    MATH  Google Scholar 

  30. G. Cherlin, Groups of small Morley rank, Ann. Math. Logic 17 (1979), 53–74.

    Article  MathSciNet  Google Scholar 

  31. L.-J. Corredor, Bad groups of finite Morley rank, J. Symbolic Logic, 54 (1989), 768–773.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Dieudonne, La Geometrie des Groupes Classiques, Springer-Verlag, 1955.

    MATH  Google Scholar 

  33. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.

    MathSciNet  MATH  Google Scholar 

  34. D. M. Goldschmidt, 2-signalizer functors on finite groups, J. Algebra 21 (1972), 321–340.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1980.

    MATH  Google Scholar 

  36. D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, 1982.

    MATH  Google Scholar 

  37. D. Gorenstein and K. Harada, Finite simple groups of low 2-rank and the families G2(g), D 24 (q), q odd, Bull. Amer. Math. Soc. 77 (1971), 829–862.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4 elements, Mem. Amer. Math. Soc. 147 (1974).

    Google Scholar 

  39. M. E. Harris, Finite groups containing an intrinsic 2-component of Chevalley type over a field of odd order, Trans. Amer. Math. Soc. 272 (1982), 1–65.

    MathSciNet  MATH  Google Scholar 

  40. B. Hartley, Simple locally finite groups, these Proceedings.

    Google Scholar 

  41. B. Hartley and G. Shute, Monomorphisms and direct limits of finite groups of Lie type, Quart. J. Math. 35 (1984), 49–71.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, Berlin-New York, 2nd edition, 1981.

    MATH  Google Scholar 

  43. S. V. Ivanov and A. Yu. Ol’shanski, Some applications of graded diagrams in combinatorial group theory, to appear.

    Google Scholar 

  44. N. Iwahori, Centralizers of involutions in finite Chevalley groups, in Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes in Math. 131, Springer-Verlag, Berlin, 1970.

    Google Scholar 

  45. O. H. Kegel, Über einfache local endliche Gruppen, Math. Z. 95 (1967), 169–195.

    Article  MathSciNet  MATH  Google Scholar 

  46. Kourovka Notebook (Unsolved Problems in Group Theory), 10th edition, Institute of Mathematics SO AN SSSR, Novosibirsk, 1986.

    Google Scholar 

  47. M. W. Liebeck and G. M. Seitz, Subgroups generated by root elements in groups of Lie type, Ann. Math. 139 (1994), 293–361.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Macintyre, On ω1-categorical theories of abelian groups, Fund. Math. 70 (1971), 253–270.

    MathSciNet  MATH  Google Scholar 

  49. A. Macintyre, On ω1-categorical theories of fields, Fund. Math. 71 (1971), 1–25.

    MathSciNet  MATH  Google Scholar 

  50. A. MacWiUiams, On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups, Trans. Amer. Math. Soc. 150 (1970), 345–408.

    MathSciNet  Google Scholar 

  51. A. Nesin, Solvable groups of finite Morley rank, J. Algebra 121 (1989), 26–39.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Nesin, Poly-separated and ω-stable nilpotent groups, J. Symbolic Logic 56 (1991), 915–931.

    MathSciNet  MATH  Google Scholar 

  53. A. Nesin, Generalized Fitting subgroup of a group of finite Morley rank, J. Symbolic Logic 56 (1991), 1391–1399.

    Article  MathSciNet  MATH  Google Scholar 

  54. B. Poizat, Une theorie de Galois imaginaire, J. Symbolic Logic 48 (1983), 1151–1170.

    Article  MathSciNet  MATH  Google Scholar 

  55. B. Poizat, Groupes Stables, Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne, France, 1987.

    MATH  Google Scholar 

  56. G. M. Seitz, Algebraic groups, these Proceedings.

    Google Scholar 

  57. R. Steinberg, Lectures on Chevalley groups, Yale University, 1967.

    Google Scholar 

  58. A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd ed., Berkeley, University of California Press, 1951.

    MATH  Google Scholar 

  59. S. Thomas, The classification of simple periodic linear groups, Arch. Math. 41 (1983), 103–116.

    Article  MATH  Google Scholar 

  60. J. H. Walter, The B-Conjecture; characterizations of Chevalley groups, Memoirs Amer. Math. Soc. 61 (1986), 1–196.

    Google Scholar 

  61. W. J. Wong, A characterization of the finite projective symplectic groups PSp4(q), Trans. Amer. Math. Soc. 139 (1969), 1–35.

    MathSciNet  MATH  Google Scholar 

  62. B. Zil’ber, Groups with categorical theories, in Fourth All-Union Symposium on Group Theory,Math. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, 1973, pp. 63–68 (in Russian).

    Google Scholar 

  63. B. Zil’ber, The structure of models of categorical theories and the problem of axiomatizability, manuscript deposited with VINITI, Dep. No. 2800–77 (in Russian).

    Google Scholar 

  64. B. ZiPber, Groups and rings whose theory is categorical, Fund. Math. 55 (1977), 173–188.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Borovik, A.V. (1995). Simple Locally Finite Groups of Finite Morley Rank and Odd Type. In: Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M. (eds) Finite and Locally Finite Groups. NATO ASI Series, vol 471. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0329-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-0329-9_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4145-4

  • Online ISBN: 978-94-011-0329-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics