Journal of Algebraic Combinatorics

, Volume 29, Issue 2, pp 195–213 | Cite as

On the order of a non-abelian representation group of a slim dense near hexagon



In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and |R|=2β, 1+NPdim(S)≤β≤1+dimV(S), where NPdim(S) is the near polygon embedding dimension of S and dimV(S) is the dimension of the universal representation module V(S) of S. Further, if β=1+NPdim(S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4.


Near polygons Non-abelian representations Generalized quadrangles Extraspecial 2-groups 


  1. 1.
    Brouwer, A.E., Cohen, A.M., Hall, J.I., Wilbrink, H.A.: Near polygons and Fischer spaces. Geom. Dedicata 49(3), 349–368 (1994) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedicata 14(2), 145–176 (1983) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cameron, P.J.: Projective and polar spaces. Available from
  4. 4.
    De Bruyn, B.: Near Polygons. Frontiers in Mathematics. Birkhäuser, Basel (2006) MATHGoogle Scholar
  5. 5.
    De Bruyn, B., Vandecasteele, P.: Near polygons with a nice chain of sub-near polygons. J. Combin. Theory Ser. A 108(2), 297–311 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    De Bruyn, B., Vandecasteele, P.: The classification of the slim dense near octagons. European J. Combin. 28(1), 410–428 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Doerk, K., Hawkes, T.: Finite Soluble Groups. de Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter & Co., Berlin (1992) MATHGoogle Scholar
  8. 8.
    Ivanov, A.A.: Non-abelian representations of geometries. Groups and combinatorics—in memory of Michio Suzuki. Adv. Stud. Pure Math. 32, 301–314 (2001) Math. Soc. Japan, Tokyo Google Scholar
  9. 9.
    Ivanov, A.A., Pasechnik, D.V., Shpectorov, S.V.: Non-abelian representations of some sporadic geometries. J. Algebra 181(2), 523–557 (1996) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics, vol. 110. Pitman (Advanced Publishing Program), Boston (1984) MATHGoogle Scholar
  11. 11.
    Ronan, M.A.: Embeddings and hyperplanes of discrete geometries. European J. Combin. 8(2), 179–185 (1987) MATHMathSciNetGoogle Scholar
  12. 12.
    Sahoo, B.K., Sastry, N.S.N.: A characterization of finite symplectic polar spaces of odd prime order. J. Combin. Theory Ser. A 114, 52–64 (2007) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shult, E., Yanushka, A.: Near n-gons and line systems. Geom. Dedicata 9(1), 1–72 (1980) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyRourkelaIndia
  2. 2.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia

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