Symmetries of biplanes
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Symmetries of biplanes Seyed Hassan Alavi1
· Ashraf Daneshkhah1 · Cheryl E. Praeger2
Received: 9 April 2020 / Revised: 5 July 2020 / Accepted: 13 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we first study biplanes D with parameters (v, k, 2), where the block size k ∈ {13, 16}. These are the smallest parameter values for which a classification is not available. We show that if k = 13, then either D is the Aschbacher biplane or its dual, or Aut(D) is a subgroup of the cyclic group of order 3. In the case where k = 16, we prove that |Aut(D)| divides 27 · 32 · 5 · 7 · 11 · 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 pointprimitive 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. Keywords Biplane · Automorphism group · Cartesian decomposition · Primitive permutation group Mathematics Subject Classification 05B05 · 05B25 · 20B25
1 Introduction A symmetric design D = (P , B) with parameters (v, k, λ) is an incidence structure consisting of a set P of v points and a set B of v blocks, each of which is a k-subset of P , such that every pair of points lies in exactly λ blocks. These conditions imply that every point is incident
Communicated by V. D. Tonchev. Dedicated to the memory of our friend and colleague Jan Saxl.
B
Seyed Hassan Alavi [email protected]; [email protected] Ashraf Daneshkhah [email protected] Cheryl E. Praeger [email protected]
1
Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran
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
123
S. H. Alavi et al. Table 1 Biplanes with small block size
Line
v
k
# Examples
References
1
7
4
1
[23]
2
11
5
1
[23]
3
16
6
3
[4,8,20]
4
37
9
4
[5]
5
56
11
5
[14,22,25,34]
6
79
13
≥2
[3,25,29]
7
121
16
Unknown
with exactly k blocks, and every pair of blocks intersects in exactly λ points. If λ = 1, then D is called a projective plane, and if λ = 2, then D is called a biplane. The aim of the paper is to study the possible symmetries of biplanes. Whereas there are several infinite families of projective planes, we currently know of only sixteen nontrivial biplanes, and they all have block size k ≤ 13, as summarised in Table 1. The column headed “# Examples” contains the number of biplanes up to isomorphism for the given parameters. In the column headed “Reference” we provide some references which give details of the constructions or classification, and in Sect. 3 we discuss briefly some of their properties. In line 7, we list the smallest set of feasible parameters for which a b
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