On a Result of Cameron and Praeger on Block-transitive Point-imprimitive t-designs

  • Michel Sebille
Conference paper

Abstract

In 1993, Cameron and Praeger proved that if G is a block-transitive point-imprimitive automorphism group of an Sλ (t, k, c2) where c =( 2 k ) − 1, k > 5, k ≠ 8, t >1, then there are two simple 2-transitive permutation groups T1 and T 2 of degree c such that one of the following holds:
  1. (i)

    G is a subgroup of the wreath product Aut(T1) ≀ Sc containing T 1 c and G projects onto a 2-transitive subgroup of Sc,

     
  2. (ii)

    T1 × T2 ≤ G ≤ Aut(T1) × Aut(T2).

     

Moreover, if (i) or (ii) holds then G acts in this way on such a design.

The purpose of this paper is to construct explicit extra-examples showing that this theorem is no longer valid for k ≤ 5 and for k = 8.

Keywords

Nickel 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W.O. Alltop, On the construction of block designs, J. Combin. Theory 1 (1966), 501–502.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Betten, R.Laue, A.Wassermann, http://btm2xl.mat.uni-bayreuth.de/ discreta/index.htmlGoogle Scholar
  3. 3.
    R.E. Block, On the orbits of collineation groups, Math. Z. 96 (1967), 33–49.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    P.J.Cameron, C.E.Praeger, Block-transitive t-designs I: point-imprimitive designs, Discrete Math. 118 (1993), 33–43.MathSciNetCrossRefGoogle Scholar
  5. 5.
    A.R. Camina, A survey of the automorphism groups of block designs, J. Combin. Designs 2 (1994), 79–100.Google Scholar
  6. 6.
    A.R. Camina, L. Di Martino, The group of automorphisms of a transitive 2 — (91,6,1) design, Geom. Dedicata 31 (1989), 151–164.MATHGoogle Scholar
  7. 7.
    A.R. Camina, L.Di Martino, Block designs on 196 points, Arch. Math. 53 (1989), 414–416.MathSciNetMATHGoogle Scholar
  8. 8.
    H. Davies, Flag-transitivity and primitivity, Discrete Math. 63 (1987), 91–93.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A. Delandtsheer, Line-primitive groups of small rank, Discrete Math. 68 (1988), 103–106.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. Delandtsheer, Line-primitive automorphism groups of finite linear spaces, European J. Combin. 10 (1989), 161–169Google Scholar
  11. 11.
    A.Delandtsheer, J.Doyen, Most block-transitive t-designs are point-primitive, Geom. Dedicata 29 (1989), 307–310.CrossRefGoogle Scholar
  12. 12.
    E.S. Kramer, S.S. Magliveras and R.Mathon, The Steiner systems S(2,4,25) with non-trivial automorphism group, Discrete Math. 77 (1989), 137–157.Google Scholar
  13. 13.
    E.S. Kramer, D.M.Mesner, t-designs on hypergraphs, Discrete Math. 15 (1976), 263–296.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    W. Nickel, A.Niemeyer, C.M. O’Keefe, T.Penttila, C.E.Praeger, The block- transitive, point imprimitive 2–(729,8,1) designs, A. pl. Algebra Engrg. Comm. Comput. 3 (1992), 47–61.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    C.M. O’Keefe, T.Penttila, C.E.Praeger, Block-transitive, point-imprimitive designs with ⋌ = 1, Discrete Math. 115 (1993), 231–244.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Michel Sebille
    • 1
  1. 1.Département de Mathématiques Campus Plaine C.P. 216Université Libre de BruxellesBruxellesBelgium

Personalised recommendations