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Designs, Codes and Cryptography

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01.05.2016

Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2/2016

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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 \(\mathrm {PSL}(2, q)\), for \(q\ne 2,3\). In particular we determine all such possible parameters \((v,k,\lambda )\) and show that there exist five isomorphic classes of such designs with \(\lambda =1,2,3,4\).

Vorheriger Artikel On automorphism groups of divisible designs acting regularly on the set of point classes

Nächster Artikel A class of five-weight cyclic codes and their weight distribution

Braić S.: Primitive symmetric designs with at most 255 points. Glas. Mat. Ser. III 45(65)(2):291–305 (2010). doi:10.3336/gm.45.2.01.

Braić S., Golemac A., Mandić J., Vučičić T.: Primitive symmetric designs with up to 2500 points. J. Comb. Des. 19(6), 463–474 (2011). doi:10.1002/jcd.20291.

Cameron P.J., Maimani H.R., Omidi G.R., Tayfeh-Rezaie B.:3-designs from \({\rm PSL}(2,q)\). Discret. Math. 306(23),3063–3073 (2006). doi:10.1016/j.disc.2005.06.041.

Camina A.R.: A survey of the automorphism groups of block designs. J. Comb. Des. 2(2), 79–100 (1994). doi:10.1002/jcd.3180020205.

Davies H.: Flag-transitivity and primitivity. Discret. Math. 63(1), 91–93 (1987). doi:10.1016/0012-365X(87)90154-3.

Dempwolff U.: Primitive rank 3 groups on symmetric designs. Des. Codes Cryptogr. 22(2), 191–207 (2001). doi:10.1023/A:1008373207617.

Dong H., Zhou S.: Alternating groups and flag-transitive \(2\text{- }(v, k,4)\) symmetric designs. J. Comb. Des. 19(6), 475–483 (2011). doi:10.1002/jcd.20294.

Dong H., Zhou S.: Affine groups and flag-transitive triplanes. Sci. China Math. 55(12), 2557–2578 (2012). doi:10.1007/s11425-012-4476-x.

Faradžev I.A., Ivanov A.A.:Distance-transitive representations of groups \(G\) with \({\rm PSL}_2(q)\unlhd G\unlhd {\rm P}\varGamma {\rm L}_2(q)\). Eur. J. Comb. 11(4), 347–356 (1990). doi:10.1016/S0195-6698(13)80137-0.

10.

Giudici M.: Maximal subgroups of almost simple groups with socle \(PSL(2, q)\). ArXiv Mathematics e-prints (2007). arXiv:math/0703685v1.

11.

Hughes D.R., Piper F.C.: Design Theory. Cambridge University Press, Cambridge (1985). doi:10.1017/CBO9780511566066.

12.

Kantor W.M.: Primitive permutation groups of odd degree, and an application to finite projective planes. J. Algebra 106(1), 15–45 (1987). doi:10.1016/0021-8693(87)90019-6.

13.

Kantor W.M., Liebler R.A.: The rank \(3\) permutation representations of the finite classical groups. Trans. Am. Math. Soc. 271(1), 1–71 (1982). doi:10.2307/1998750.

14.

Lander E.S.: Symmetric Designs: An Algebraic Approach. London Mathematical Society Lecture Note Series, vol. 74. Cambridge University Press, Cambridge (1983). doi:10.1017/CBO9780511662164.

15.

Law M., Praeger C.E., Reichard S.: Flag-transitive symmetric \(2\text{- }(96,20,4)\)-designs. J. Comb. Theory Ser. A 116(5), 1009–1022 (2009). doi:10.1016/j.jcta.2008.12.005.

16.

O’Reilly-Regueiro E.: Flag-transitive symmetric designs. Ph.D. thesis, University of London (2003).

17.

O’Reilly-Regueiro E.: Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle. Eur. J. Comb. 26(5), 577–584 (2005). doi:10.1016/j.ejc.2004.05.003.

18.

O’Reilly-Regueiro E.: On primitivity and reduction for flag-transitive symmetric designs. J. Comb. Theory Ser. A 109(1), 135–148 (2005). doi:10.1016/j.jcta.2004.08.002.

19.

O’Reilly-Regueiro E.: Biplanes with flag-transitive automorphism groups of almost simple type, with classical socle. J. Algebraic Comb. 26(4), 529–552 (2007). doi:10.1007/s10801-007-0070-7.

20.

O’Reilly-Regueiro E.: Biplanes with flag-transitive automorphism groups of almost simple type, with exceptional socle of Lie type. J. Algebraic Comb. 27(4), 479–491 (2008). doi:10.1007/s10801-007-0098-8.

21.

Praeger C.E., Zhou S.: Imprimitive flag-transitive symmetric designs. J. Comb. Theory Ser. A 113(7), 1381–1395 (2006). doi:10.1016/j.jcta.2005.12.006.

22.

Seitz G.M.: Flag-transitive subgroups of Chevalley groups. Ann. Math. 2(97), 27–56 (1973).

23.

Tian D., Zhou S.: Flag-transitive point-primitive symmetric \((v, k,\lambda )\) designs with \(\lambda \) at most 100. J. Comb. Des. 21(4), 127–141 (2013). doi:10.1002/jcd.21337.

24.

The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.6.4 (2013).

25.

Zhou S., Dong H.: Sporadic groups and flag-transitive triplanes. Sci. China Ser. A 52(2), 394–400 (2009). doi:10.1007/s11425-009-0011-0.

26.

Zhou S., Dong H.: Alternating groups and flag-transitive triplanes. Des. Codes Cryptogr. 57(2), 117–126 (2010). doi:10.1007/s10623-009-9355-2.

27.

Zhou S., Dong H.: Exceptional groups of Lie type and flag-transitive triplanes. Sci. China Math. 53(2), 447–456 (2010). doi:10.1007/s11425-009-0051-5.

28.

Zhou S., Tian D.: Flag-transitive point-primitive 2-\((v, k,4)\) symmetric designs and two dimensional classical groups. Appl. Math. J. Chinese Univ. Ser. B 26(3), 334–341 (2011). doi:10.1007/s11766-011-2702-x.

29.

Zhou S., Dong H., Fang W.: Finite classical groups and flag-transitive triplanes. Discret. Math. 309(16), 5183–5195 (2009). doi:10.1016/j.disc.2009.04.005.

Titel: Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups
Publikationsdatum: 01.05.2016
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2/2016
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-015-0055-9

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Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups

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