Non-symmetric 2-designs admitting a two-dimensional projective linear group

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Abstract

This paper is a contribution to the study of non-symmetric 2-designs admitting a flag-transitive automorphism group. We prove that if \(\mathcal {D}\) is a non-trivial non-symmetric 2-\((v, k, \lambda )\) design with \((r,\lambda ) = 1\) and \(G=PSL(2,q)\) acts flag-transitively on \(\mathcal {D}\), then up to isomorphism \(\mathcal {D}\) is a unique Witt-Bose-Shrikhande space, a unique 2-(6, 3, 2) design, a unique 2-(8, 4, 3) design, a unique 2-(10, 6, 5) design, or a unique 2-(28, 7, 2) design.

Keywords

2-Design Non-symmetric Automorphism Flag-transitive Projective linear group 

Mathematics Subject Classification

05B05 05B25 20B25 

Notes

Acknowledgements

The authors would like to thank referees for providing us helpful and constructive comments and suggestions, which led to the improvement of the article. This work is supported by the National Natural Science Foundation of China (Grant No. 11471123).

References

  1. 1.
    Alavi S.H., Bayat M., Daneshkhah A.: Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups. Des. Codes Cryptogr. 79(2), 337–351 (2016).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alavi S.H., Burness T.C.: Large subgroups of simple groups. J. Algebra 421, 187–233 (2015).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Assmus E.F., Prince A.R.: Biplanes and near biplanes. J. Geom. 40(1), 1–14 (1991).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Biliotti M., Montinaro A.: On flag-transitive symmetric designs of affine type. J. Combin. Des. 25(2), 85–97 (2017).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bray J.N., Holt D.F., Roney-Dougal C.M.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 407. Cambridge University Press, Cambridge (2013)Google Scholar
  6. 6.
    Buekenhout F., Delandtsheer A., Doyen J.: Finite linear spaces with flag-transitive group. J. Combin. Theory Ser. A 49, 268–293 (1988).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Buekenhout F., Delandtsheer A., Doyen J., Kleidman P.B., Liebeck M., Saxl J.: Linear spaces with flag-transitive automorphism groups. Geom. Dedicata 36, 89–94 (1990).MathSciNetMATHGoogle Scholar
  8. 8.
    Colbourn C.J., Dinitz J.H.: The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL (2007).MATHGoogle Scholar
  9. 9.
    Davies H.: Flag-transitivity and primitivity. Discret. Math. 63, 91–93 (1987).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Delandtsheer A.: Flag-transitive finite simple groups. Arch. Math. 47, 395–400 (1986).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dembowski P.: Finite Geometries. Springer, New York (1968).CrossRefMATHGoogle Scholar
  12. 12.
    Faradzev I.A., Ivanov A.A.: Distance-transitive representations of groups \(G\) with \(PSL(2, q)\unlhd G\le P\Gamma L(2, q)\). Eur. J. Combin. 11, 347–356 (1990).CrossRefMATHGoogle Scholar
  13. 13.
    Passman D.S.: Permutation Groups. Benjamin, New York (1968).MATHGoogle Scholar
  14. 14.
    The GAP Group: GAPC Groups, Algorithms, and Programming, Version 4.4; (2004). http://www.gapsystem.org.
  15. 15.
    Wielandt H.: Finite Permutation Groups. Academic Press, New York (1964).MATHGoogle Scholar
  16. 16.
    Zhan X.Q., Zhou S.L.: Flag-transitive non-symmetric 2-designs with \((r,\lambda )=1\) and sporadic socle. Des. Codes Cryptogr. 81(3), 481–487 (2016).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zhou S.L., Wang Y.J.: Flag-transitive non-symmetric 2-designs with \((r,\lambda )=1\) and alternating socle. Electron. J. Combin. 22(2), #P2.6 (2015).Google Scholar
  18. 18.
    Zhu Y., Guan H.Y., Zhou S.L.: Flag-transitive 2-\((v, k,\lambda )\) symmetric designs with \((k,\lambda )=1\) and alternating socle. Front. Math. China 10(6), 1483–1496 (2015).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zieschang P.H.: Flag transitive automorphism groups of 2-designs with \((r,\lambda )=1\). J. Algebra 118, 265–275 (1988).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceEast China JiaoTong UniversityNanchangPeople’s Republic of China
  2. 2.School of MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China

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