Glasnik Matematicki, Vol. 40, No.1 (2005), 1-11.


Mario Essert and Ljubo Marangunic

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lucica 1, 10000 Zagreb, Croatia

Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia

Abstract.   The aim of this article is to prove that exactly four biplanes with parameters (56,11,2) admit a fixed-point-free action of an involutory automorphism. These are: Hall's biplane B20, Salwach and Mezzaroba's biplane B22, Denniston's biplane B24 and Denniston's biplane B26.

2000 Mathematics Subject Classification.   05B05.

Key words and phrases.   Biplane, design, automorphism group, orbit, orbit structure.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.1.01


  1. V. Cepulic, A Symmetric Block Design (45,12,3) with automorphisms of Order 5, Ars Combinatoria 37 (1994), 33-48.

  2. V. Cepulic and M. Essert, Biplanes and their automorphisms, Studia Sci. Math. Hungar. 24 (1989), 437-446.

  3. V. Cepulic and M. Essert, Biplanes (56,11,2) with involutory automorphism fixing 14 points, Glasnik Mat. 31(51) (1996), 25-38.

  4. V. Cepulic and Lj. Marangunic, Biplanes (56,11,2) with nontrivial automorphisms, preprint.

  5. Z. Janko and Tran van Trung, A new biplane of order 9 with a small automorphism group, Math. Institut Heidelberg, Heidelberg, 1985.

  6. Z. Janko and Tran van Trung, Construction of a new symmetric block design for (78,22,6) with the help of tactical decompositions, J. Combin. Theory Ser. A 40 (1985), 451-455.

  7. Lj. Marangunic, Biplanes (56,11,2) with involutory collineation fixing 6 points, J. Combin. Theory Ser. A 54 (1990), 149-163.

  8. Th. C. J. Salwach and J. A. Mezzaroba, The four known biplanes with k = 11, Internat. J. Math. Math. Sci. 2 (1979), 251-260.

Glasnik Matematicki Home Page