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2 results

  • 5 - Fischer spaces

  • Andries E. Brouwer, Technische Universiteit Eindhoven, The Netherlands, H. Van Maldeghem, Universiteit Gent, Belgium
  • Book:
    Strongly Regular Graphs
    Published online:
    06 January 2022
    Print publication:
    13 January 2022, pp 139-154

  • Summary

    Fischer classified the groups generated by a conjugacy class 𝐷 of 3-transpositions (involutions such that the product of any two has order at most 3) and discovered three new sporadic groups that bear his name. These groups are rank 3 groups: 𝐷 carries in a natural way the structure of a geometry with lines of length 3 and the structure of a rank 3 graph. We review some properties of these geometries, called Fischer spaces, and mention Fischer’s group-theoretic classification. We discuss the examples and give detailed parameter information. We briefly discuss cotriangular spaces and Shult’s classification, and Hall’s classification of copolar spaces. Finally we classify with full proofs all lax embeddings of (finite) symplectic copolar spaces in projective space. We use some specific such embedded copolar spaces to construct two rank 3 graphs of affine type, one with 2401 vertices, the other with 6561 vertices.

  • ULTRAHOMOGENEOUS SEMILINEAR SPACES

  • ALICE DEVILLERS
  • Journal:
    Proceedings of the London Mathematical Society / Volume 84 / Issue 1 / March 2002
    Published online by Cambridge University Press:
    13 February 2002, pp. 35-58
    Print publication:
    March 2002

  • A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extends a result of A. Gardiner on graphs and a result of A. Devillers and J. Doyen on linear spaces.

    2000 Mathematical Subject Classification: 05B25, 51E14, 20B25.