Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FLAG-TRANSITIVE POINT-PRIMITIVE SYMMETRIC DESIGNS AND THREE DIMENSIONAL PROJECTIVE SPECIAL UNITARY GROUPS

  • Daneshkhah, Ashraf (Department of Mathematics Faculty of Science Bu-Ali Sina University) ;
  • Zarin, Sheyda Zang (Department of Mathematics Faculty of Science Bu-Ali Sina University)
  • Received : 2016.08.24
  • Accepted : 2016.12.26
  • Published : 2017.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

The main aim of this article is to study symmetric (v, k, ${\lambda}$) designs admitting a flag-transitive and point-primitive automorphism group G whose socle is PSU(3, q). We indeed show that such designs must be complete.

Keywords

References

  1. S. H. Alavi and M. Bayat, Flag-transitive point-primitive symmetric designs and three dimensional projective special linear groups, Bull. Iranian Math. Soc. 42 (2016), no. 1, 201-221.
  2. S. H. Alavi, M. Bayat, and A. Daneshkhah, Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups, Des. Codes Cryptogr. 79 (2016), no. 2, 337-351. https://doi.org/10.1007/s10623-015-0055-9
  3. J. N. Bray, D. F. Holt, and C. M. Roney-Dougal, The maximal subgroups of the low- dimensional finite classical groups, volume 407 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2013.
  4. P. J. Cameron and C. E. Praeger, Constructing flag-transitive, point-imprimitive designs, J. Algebraic Combin. 43 (2016), no. 4, 755-769. https://doi.org/10.1007/s10801-015-0591-4
  5. J. D. Dixon and B. Mortimer, Permutation groups, volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996.
  6. H. Dong and S. Zhou, Alternating groups and flag-transitive 2-($\nu$, ${\kappa}$, 4) symmetric designs, J. Combin. Des. 19 (2011), no. 6, 475-483. https://doi.org/10.1002/jcd.20294
  7. H. Dong, Ane groups and flag-transitive triplanes, Sci. China Math. 55 (2012), no. 12, 2557-2578. https://doi.org/10.1007/s11425-012-4476-x
  8. The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.7.9, 2015.
  9. R. W. Hartley, Determination of the ternary collineation groups whose coeffcients lie in the GF($2^n$), Ann. of Math. (2) 27 (1925), no. 2, 140-158. https://doi.org/10.2307/1967970
  10. D. R. Hughes and F. C. Piper, Design Theory, Cambridge University Press, Cambridge, 1985.
  11. W. M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra 106 (1987), no. 1, 15-45. https://doi.org/10.1016/0021-8693(87)90019-6
  12. P. B. Kleidman, The subgroup structure of some finite simple groups, Ph.D. Thesis, University of Cambridge, 1987.
  13. E. S. Lander, Symmetric designs: an algebraic approach, volume 74 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1983.
  14. M. Law, C. E. Praeger, and S. Reichard, Flag-transitive symmetric 2-(96, 20, 4)-designs, J. Combin. Theory Ser. A 116 (2009), no. 5, 1009-1022. https://doi.org/10.1016/j.jcta.2008.12.005
  15. M. W. Liebeck, J. Saxl, and G. M. Seitz, On the overgroups of irreducible subgroups of the finite classical groups, Proc. London Math. Soc. (3) 55 (1987), no. 3, 507-537.
  16. H. H. Mitchell, Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), no. 2, 207-242. https://doi.org/10.1090/S0002-9947-1911-1500887-3
  17. E. O'Reilly-Regueiro, Flag-transitive symmetric designs, PhD thesis, University of London, 2003.
  18. E. O'Reilly-Regueiro, Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle, European J. Combin. 26 (2005), no. 5, 577-584. https://doi.org/10.1016/j.ejc.2004.05.003
  19. E. O'Reilly-Regueiro, On primitivity and reduction for flag-transitive symmetric designs, J. Combin. Theory Ser. A 109 (2005), no. 1, 135-148. https://doi.org/10.1016/j.jcta.2004.08.002
  20. E. O'Reilly-Regueiro, Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle, J. Algebraic Combin. 26 (2007), no. 4, 529-552. https://doi.org/10.1007/s10801-007-0070-7
  21. E. O'Reilly-Regueiro, Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type, J. Algebraic Combin. 27 (2008), no. 4, 479-491. https://doi.org/10.1007/s10801-007-0098-8
  22. C. E. Praeger and S. Zhou, Imprimitive flag-transitive symmetric designs, J. Combin. Theory Ser. A 113 (2006), no. 7, 1381-1395. https://doi.org/10.1016/j.jcta.2005.12.006
  23. G. M. Seitz, Flag-transitive subgroups of Chevalley groups, Ann. of Math. (2) 97 (1973), 27-56. https://doi.org/10.2307/1970876
  24. D. Tian and S. Zhou, Flag-transitive point-primitive symmetric ($\nu$, ${\kappa}$, $\lambda$) designs with  at most 100, J. Combin. Des. 21 (2013), no. 4, 127-141. https://doi.org/10.1002/jcd.21337
  25. D. Tian, Flag-transitive 2- ($\nu$, ${\kappa}$, $\lambda$) symmetric designs with sporadic socle, J. Combin. Des. 23 (2015), no. 4, 140-150. https://doi.org/10.1002/jcd.21385
  26. D. Tian, Classi cation of flag-transitive primitive symmetric ($\nu$, ${\kappa}$, $\lambda$) designs with PSL(2, q) as socle, J. Math. Res. Appl. 36 (2016), no. 2, 127-139.
  27. S. Zhou and H. Dong, Sporadic groups and flag-transitive triplanes, Sci. China Ser. A 52 (2009), no. 2, 394-400. https://doi.org/10.1007/s11425-009-0011-0
  28. S. Zhou, Alternating groups and flag-transitive triplanes, Des. Codes Cryptogr. 57 (2010), no. 2, 117-126. https://doi.org/10.1007/s10623-009-9355-2
  29. S. Zhou, Exceptional groups of Lie type and flag-transitive triplanes, Sci. China Math. 53 (2010), no. 2, 447-456. https://doi.org/10.1007/s11425-009-0051-5
  30. S. Zhou, H. Dong, and W. Fang, Finite classical groups and flag-transitive triplanes, Discrete Math. 309 (2009), no. 16, 5183-5195. https://doi.org/10.1016/j.disc.2009.04.005
  31. S. Zhou and D. Tian, Flag-transitive point-primitive 2-($\nu$, ${\kappa}$, 4) symmetric designs and two dimensional classical groups, Appl. Math. J. Chinese Univ. Ser. B 26 (2011), no. 3, 334-341. https://doi.org/10.1007/s11766-011-2702-x