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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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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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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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