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2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

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Abstract

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, BAS, P.O. Box 323, 5000, V. Tarnovo, Bulgaria

    Svetlana Topalova & Stela Zhelezova

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  1. Svetlana Topalova
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  2. Stela Zhelezova
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Correspondence to Svetlana Topalova.

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This work was partially supported by the Bulgarian National Science Fund, Contract No MM 1405. Part of the results were announced at the Thirty Seventh Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 2008, Bulgaria.

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Topalova, S., Zhelezova, S. 2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2). Graphs and Combinatorics 26, 727–735 (2010). https://doi.org/10.1007/s00373-010-0943-8

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  • DOI: https://doi.org/10.1007/s00373-010-0943-8

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