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Designs, Codes and Cryptography

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07-08-2020

Symmetries of biplanes

Authors: Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E. Praeger

Published in: Designs, Codes and Cryptography | Issue 11/2020

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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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Title: Symmetries of biplanes
Authors: Seyed Hassan Alavi
Ashraf Daneshkhah
Cheryl E. Praeger
Publication date: 07-08-2020
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 11/2020
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-020-00784-1

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