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Symmetries of biplanes

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Abstract

In this paper, we first study biplanes \(\mathcal {D}\) with parameters (v, k, 2), where the block size \(k\in \{13,16\}\). These are the smallest parameter values for which a classification is not available. We show that if \(k=13\), then either \(\mathcal {D}\) is the Aschbacher biplane or its dual, or \(\mathbf {Aut}(\mathcal {D})\) is a subgroup of the cyclic group of order 3. In the case where \(k=16\), we prove that \(|\mathbf {Aut}(\mathcal {D})|\) divides \(2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13\). We also provide an example of a biplane with parameters (16, 6, 2) with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.

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References

  1. Andreescu T., Andrica D.: Quadratic Diophantine Equations. Developments in MathematicsSpringer, New York (2015). https://doi.org/10.1007/978-0-387-54109-9. With a foreword by Preda Mihăilescu.

    Book  MATH  Google Scholar 

  2. Arasu K.T., Mavron V.C.: Biplanes and Singer Groups. In: Jungnickel D., Vanstone S.A. (eds.) Coding Theory, Design Theory, Group Theory (Burlington, VT, 1990), pp. 111–119. Wiley, New York (1993).

    Google Scholar 

  3. Aschbacher M.: On collineation groups of symmetric block designs. J. Combinatorial Theory Ser. A 11, 272–281 (1971). https://doi.org/10.1016/0097-3165(71)90054-9.

    Article  MathSciNet  MATH  Google Scholar 

  4. Assmus Jr. E.F., Salwach C.J.: The \((16,\,6,\,2)\) designs. Internat. J. Math. Math. Sci. 2(2), 261–281 (1979). https://doi.org/10.1155/S0161171279000247.

    Article  MathSciNet  MATH  Google Scholar 

  5. Assmus Jr., E.F., Mezzaroba, J.A., Salwach, C.J.: Planes and biplanes. In: Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976), NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci. 31, 205–212 (1977)

  6. Betten A., Delandtsheer A., Niemeyer A.C., Praeger C.E.: On a theorem of Wielandt for finite primitive permutation groups. J. Group Theory 6(4), 415–420 (2003). https://doi.org/10.1515/jgth.2003.029.

    Article  MathSciNet  MATH  Google Scholar 

  7. Block R.E.: On the orbits of collineation groups. Math. Z. 96, 33–49 (1967).

    Article  MathSciNet  Google Scholar 

  8. Burau W.: Über die zur Kummerkonfiguration analogen Schemata von 16 Punkten und 16 Blöcken und ihre Gruppen. Abh. Math. Sem. Univ. Hamburg 26, 129–144 (1963). https://doi.org/10.1007/BF02992783.

    Article  MathSciNet  MATH  Google Scholar 

  9. Camina A.R.: Projective planes with a transitive automorphism group. Innov. Incidence Geom. 1, 191–196 (2005).

    Article  MathSciNet  Google Scholar 

  10. Camina A.R., Neumann P.M., Praeger C.E.: Alternating groups acting on finite linear spaces. Proc. Lond. Math. Soc. (3) 87(1), 29–53 (2003). https://doi.org/10.1112/S0024611503014060.

    Article  MathSciNet  MATH  Google Scholar 

  11. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985). Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray

  12. Denniston R.H.F.: On biplanes with \(56\) points. Ars Combin. 9, 167–179 (1980).

    MathSciNet  MATH  Google Scholar 

  13. Dixon J.D., Mortimer B.: Permutation groups. In: Dixon J.D., Mortimer B. (eds.) Graduate Texts in Mathematics. Springer, New York (1996).

    Google Scholar 

  14. Essert M., Marangunić L.: Biplanes \((56,11,2)\) with a fixed-point-free involutory automorphism. Glas. Mat. Ser. III 40(60)(1), 1–11 (2005). https://doi.org/10.3336/gm.40.1.01.

    Article  MathSciNet  MATH  Google Scholar 

  15. Gill N.: Transitive projective planes. Adv. Geom. 7(4), 475–528 (2007). https://doi.org/10.1515/ADVGEOM.2007.030.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gill N.: Transitive projective planes and insoluble groups. Trans. Am. Math. Soc. 368(5), 3017–3057 (2016). https://doi.org/10.1090/tran/6366.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gorenstein D.: Finite Groups, 2nd edn. Chelsea Publishing Co., New York (1980).

    MATH  Google Scholar 

  18. Hall Jr. M., Lane R., Wales D.: Designs derived from permutation groups. J. Combin. Theory 8, 12–22 (1970).

    Article  MathSciNet  Google Scholar 

  19. Hughes D., Piper F.: Design Theory. Up (Methuen). Cambridge University Press, Cambridge (1988).

    Google Scholar 

  20. Husain Q.M.: On the totality of the solutions for the symmetrical incomplete block designs: \(\lambda =2, k=5\) or \(6\). Sankhyā 7, 204–208 (1945).

    MathSciNet  MATH  Google Scholar 

  21. Hussain Q.M.: Symmetrical incomplete block designs with \(\lambda =2\), \(k=8\) or \(9\). Bull. Calcutta Math. Soc. 37, 115–123 (1945).

    MathSciNet  Google Scholar 

  22. Janko Z., Trung T.V.: A new biplane of order \(9\) with a small automorphism group. J. Combin. Theory Ser. A 42(2), 305–309 (1986). https://doi.org/10.1016/0097-3165(86)90103-2.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kantor W.M.: \(2\)-transitive symmetric designs. Trans. Amer. Math. Soc. 146, 1–28 (1969). https://doi.org/10.2307/1995157.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Kaski P., Östergård P.R.J.: There are exactly five biplanes with \(k=11\). J. Combin. Des. 16(2), 117–127 (2008). https://doi.org/10.1002/jcd.20145.

    Article  MathSciNet  MATH  Google Scholar 

  26. Key, J.D., Tonchev, V.D.: Computational results for the known biplanes of order \(9\). In: Geometry, combinatorial designs and related structures (Spetses, 1996), London Math. Soc. Lecture Note Ser., vol. 245, pp. 113–122. Cambridge Univ. Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511526114.011

  27. Lander, E.S.: Symmetric designs: an algebraic approach. In: London Mathematical Society Lecture Note Series, vol. 74. Cambridge University Press, Cambridge (1983). https://doi.org/10.1017/CBO9780511662164

  28. Liebeck M.W., Praeger C.E., Saxl J.: On the O’Nan–Scott theorem for finite primitive permutation groups. J. Austral. Math. Soc. Ser. A 44(3), 389–396 (1988).

    Article  MathSciNet  Google Scholar 

  29. Marangunić L.: Biplanes \((79,13,2)\) with involutory automorphism. J. Combin. Theory Ser. A 61(1), 36–49 (1992). https://doi.org/10.1016/0097-3165(92)90051-U.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Praeger C.E.: The inclusion problem for finite primitive permutation groups. Proc. Lond. Math. Soc. (3) 60(1), 68–88 (1990). https://doi.org/10.1112/plms/s3-60.1.68.

    Article  MathSciNet  MATH  Google Scholar 

  32. Praeger, C.E., Schneider, C.: Permutation groups and Cartesian decompositions. In: London Mathematical Society Lecture Note Series, vol. 449. Cambridge University Press, Cambridge (2018). https://doi.org/10.1017/9781139194006

  33. Salwach C.J., Mezzaroba J.A.: The four known biplanes with \(k=9\). J. Combin. Theory Series A 24, 141–145 (1978).

    Article  Google Scholar 

  34. Salwach C.J., Mezzaroba J.A.: The four known biplanes with \(k=11\). Internat. J. Math. Math. Sci. 2(2), 251–260 (1979). https://doi.org/10.1155/S0161171279000235.

    Article  MathSciNet  MATH  Google Scholar 

  35. The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.7.9 (2015). http://www.gap-system.org

  36. Wong W.J.: Determination of a class of primitive permutation groups. Math. Z. 99, 235–246 (1967). https://doi.org/10.1007/BF01112454.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work in the paper forms part of Australian Research Council Discovery Project Grant DP200100080. All authors thank the referees for a careful reading of the paper. The first and second authors are also grateful to Cheryl E. Praeger and Alice Devillers for supporting their visit to The University of Western Australia during July-September 2019.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran

    Seyed Hassan Alavi & Ashraf Daneshkhah

  2. Centre for the Mathematics of Symmetry and Computation, Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Cheryl E. Praeger

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  1. Seyed Hassan Alavi
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  2. Ashraf Daneshkhah
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  3. Cheryl E. Praeger
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Correspondence to Seyed Hassan Alavi.

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Alavi, S.H., Daneshkhah, A. & Praeger, C.E. Symmetries of biplanes. Des. Codes Cryptogr. 88, 2337–2359 (2020). https://doi.org/10.1007/s10623-020-00784-1

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