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Journal of Algebraic Combinatorics

, Volume 16, Issue 2, pp 111–150 | Cite as

More on Geometries of the Fischer Group Fi22

  • A.A. Ivanov
  • C. Wiedorn
Article

Abstract

We give a new, purely combinatorial characterization of geometries \(\varepsilon \) with diagram Open image in new window identifying each under some “natural” conditions—but not assuming any group action a priori—with one of the two geometries \(\mathcal{E}(Fi_{22} )\) and \(\mathcal{E}(3 \cdot Fi_{22} )\) related to the Fischer 3-transposition group Fi22 and its non-split central extension 3 · Fi22, respectively. As a by-product we improve the known characterization of the c-extended dual polar spaces for Fi22 and 3 · Fi22 and of the truncation of the c-extended 6-dimensional unitary polar space.

Fischer group diagram geometry extended building 

References

  1. 1.
    J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.zbMATHGoogle Scholar
  2. 2.
    H. Cuypers, “Finite locally generalized quadrangles with affine planes,” Europ. J. Comb. 13 (1992), 439–453.MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Hall and S.V. Shpectorov, “Rank 3 P-geometries,” Geom. Dedicata. 82 (2000), 139–169.MathSciNetCrossRefGoogle Scholar
  4. 4.
    A.A. Ivanov, S.A. Linton, K. Lux, J. Saxl, and L.H. Soicher, “Distance-transitive representations of the sporadic groups,” Comm. Algebra 23(9) (1995), 3379–3427.MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.A. Ivanov, D.V. Pasechnik, and S.V. Shpectorov, “Extended F4-buildings and the Baby Monster,” Invent. Math. 144 (2001), 399–433.MathSciNetCrossRefGoogle Scholar
  6. 6.
    A.A. Ivanov and S.V. Shpectorov, “The flag-transitive tilde and Petersen-type geometrics are all known,” Bull. Amer. Math. Soc. New Ser. 31(2) (1994), 172–184.CrossRefGoogle Scholar
  7. 7.
    A.A. Ivanov, “On geometries of the Fischer groups,” Eur. J. Comb. 16(2) (1995), 163–183.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A.A. Ivanov, Geometry of Sporadic Groups I. Peterson and Tilde Geometries, Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
  9. 9.
    T. Meixner, “Some polar towers,” European J. Combin. 12 (1991), 397–451.MathSciNetCrossRefGoogle Scholar
  10. 10.
    D.V. Pasechnik, “Geometric characterization of the sporadic groups Fi22, Fi23, and Fi24*,” J. Comb. Th. (A) 68 (1994), 100–114.CrossRefGoogle Scholar
  11. 11.
    D.V. Pasechnik, “Extended polar spaces of rank at least 3,” J. Comb. Th. (A) 72(2) (1995), 232–242.CrossRefGoogle Scholar
  12. 12.
    A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.zbMATHGoogle Scholar
  13. 13.
    M. Ronan, “Embeddings and hyperplanes of discrete geometries,” European J. Combin. 8 (1987), 179–185.MathSciNetCrossRefGoogle Scholar
  14. 14.
    G. Seitz, “Flag-transitive subgroups of Chevalley groups,” Ann. Math. 97 (1973), 27–56.MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, 386, Springer, Berlin, 1974.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • A.A. Ivanov
    • 1
  • C. Wiedorn
    • 2
  1. 1.Department of MathematicsImperial CollegeLondonUK
  2. 2.Department of MathematicsUniversity of BirminghamBirminghamUK

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