Designs, Codes and Cryptography

, Volume 47, Issue 1–3, pp 99–111 | Cite as

Classification of line-transitive point-imprimitive linear spaces with line size at most 12



In this paper we complete a classification of finite linear spaces \({\mathcal{S}}\) with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders 4, 7, 9 and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.


Linear spaces Line-transitive Point-imprimitive 

AMS Subject Classifications

05B05 05B25 20B25 


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  1. 1.
    Betten A, Praeger CE. Searching for line-transitive point-imprimitive linear spaces with a grid structure, in preparationGoogle Scholar
  2. 2.
    Betten A, Cresp G, Praeger CE. Line-transitive point-imprimitive linear spaces: the grid case, submittedGoogle Scholar
  3. 3.
    Betten A, Delandtsheer A, Law M, Niemeyer AC, Praeger CE, Zhou S Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight, submitted for publication. (available at Scholar
  4. 4.
    Camina AR, Mischke S (1996) Line-transitive automorphism groups of linear spaces. Electron J Combin 3(1): Research Paper 3, 16 pp. (electronic)Google Scholar
  5. 5.
    Camina AR and Praeger CE (1993). Line-transitive automorphism groups of linear spaces. Bull London Math Soc 25(4): 309–315 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Camina AR and Siemons J (1989). Block transitive automorphism groups of 2 − (υ,k,1) block designs. J Combin Theory Ser A 51: 268–276 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Conway JH, Curtis RT, Norton SP, Parker RA, Wilson RA (1985) Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, EynshamGoogle Scholar
  8. 8.
    Cresp G (2001) Searching for line-transitive, point-imprimitive, linear spaces. Honours Dissertation, The University of Western Australia, Perth. (available at Scholar
  9. 9.
    Delandtsheer A and Doyen J (1989). Most block-transitive t-designs are point-primitive. Geom Dedicata 29(3): 307–310 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dixon JD and Mortimer B (1996). Permutation groups. Springer-Verlag, New York MATHGoogle Scholar
  11. 11.
    The GAP Group (2004) GAP – Groups, Algorithms, and Programming, Version 4.4; ( Scholar
  12. 12.
    Lam CWH, Thiel L and Swiercz S (1989). The nonexistence of finite projective planes of order 10. Canad J Math 41(6): 1117–1123 MATHMathSciNetGoogle Scholar
  13. 13.
    Nickel W, Niemeyer AC, O’Keefe CM, Penttila T and Praeger CE (1992). The block-transitive, point-imprimitive 2-(729,8,1) designs. Appl Algebra Engrg Comm Comput 3: 47–61 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    O’Keefe CM, Penttila T and Praeger CE (1993). Block-transitive, point-imprimitive designs with λ = 1. Disc Math 115: 231–244 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Praeger CE and Tuan ND (2003). Inequalities involving the Delandtsheer-Doyen parameters for finite line-transitive linear spaces. J Combin Theory Ser A 102(1): 268–276 CrossRefMathSciNetGoogle Scholar
  16. 16.
    Stoichev SD and Tonchev VD (1988). The automorphism groups of the known 2-(91,6,1) designs. C. R. Acad Bulgare Sci 41(4): 15–16 MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of Western AustraliaCrawleyAustralia
  2. 2.Department of MathematicsShantou UniversityShantouP.R. China

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