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Parallelisms of PG(3,5) with automorphisms of order 13

Parallelisms of PG(3,5) with automorphisms of order 13. Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS,Bulgaria. Parallelisms of PG(3,5) with automorphisms of order 13. Introduction History PG(3,5) and related 2-designs Construction Results.

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Parallelisms of PG(3,5) with automorphisms of order 13

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  1. Parallelisms of PG(3,5) with automorphisms of order 13 Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS,Bulgaria

  2. Parallelisms ofPG(3,5) with automorphisms of order 13 • Introduction • History • PG(3,5) and related 2-designs • Construction • Results

  3. Parallelisms ofPG(3,5) with automorphisms of order 13 Introduction • t-spread in PG(n,q) - a set of distinct t-dimensional subspaces which partition the point set. • t-parallelism in PG(n,q) – a partition of the set of t-dimensional subspaces by t-spreads. • Spread, parallelism ≡ line spread, line parallelism ≡ 1-spread, 1-parallelism

  4. Parallelisms ofPG(3,5) with automorphisms of order 13 Introduction • Isomorphicparallelisms – exists an automorphism of PG(n,q) which maps each spread of the first parallelism to a spread of the second one. • Automorphism group of the parallelism – maps each spread of the parallelism to a spread of the same parallelism. • Transitiveparallelism – it has an automorphism group which is transitive on the spreads.

  5. Parallelisms ofPG(3,5) with automorphisms of order 13 Introduction • Regulus – a set R of q+1 mutually skew lines – any line intersecting three elements of R intersects all elements of R. • Regular spread – for every three spread lines, the unique regulus determined by them is a subset of the spread. • Regularparallelism– all its spreads are regular.

  6. Parallelisms ofPG(3,5) with automorphisms of order 13 • 2-design: • V– finite set of vpoints • B – finite collection of bblocks: k-element subsets of V • D = (V, B) – 2-(v,k,λ)design if any 2-subset of V is in λ blocks of B. • Parallel class – a partition of the point set by blocks. • Resolution – a partition of the collection of blocks by parallel classes. Introduction

  7. Parallelisms ofPG(3,5) with automorphisms of order 13 History General constructions of parallelisms: • PG(n,2) – Zaicev, G., Zinoviev, V., Semakov, N., Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-errorcorrecting codes, 1971. • – Baker, R., Partitioning the planes of AG2m(2) into 2-designs, 1976. • PG(2n-1,q) – Beutelspacher, A., On parallelisms in finite projective spaces, 1974.

  8. Parallelisms ofPG(3,5) with automorphisms of order 13 History ParallelismsinPG(3,q): • Denniston, R., Packings of PG(3,q), 1973. • Penttila, T. and Williams, B., Regular packings of PG(3,q), 1998. • Johnson, N., Combinatorics of Spreads and Parallelisms, 2010.

  9. Parallelisms ofPG(3,5) with automorphisms of order 13 History • Computer aided classifications: • PG(3,3)– with some group of automorphisms byPrince, 1997. • PG(3,4) – with automorphisms of orders 7 and 5 by us, 2009, 2013. • PG(3,5) – classification of cyclic parallelisms byPrince, 1998.

  10. Parallelisms ofPG(3,5) with automorphisms of order 13 PG(3,5) and related2-designs • The incidence of the pointsand t-dimensional subspaces of PG(n,q) defines a 2-design (D). points of D blocks of D resolutions of D points of PG(3,5) lines of PG(3,5) parallelisms of PG(3,5) 2-(156,6,1) design

  11. Parallelisms ofPG(3,5) with automorphisms of order 13 PG(3,5) and related2-designs Parallelisms ofPG(3,5) 31 spreads with26lines

  12. Parallelisms ofPG(3,5) with automorphisms of order 13 PG(3,5) and related2-designs PG(3,5) points, lines. • G – group of automorphismsof PG(3,5): |G| =29 . 32 . 56 . 13 . 31 Gi – subgroup oforderi. G13 – GAP –http://www.gap-system.org • G – group of automorphisms of the related toPG(3,5)designs.

  13. Parallelisms ofPG(3,5) with automorphisms of order 13 Construction • Sylow subgroup of order 13 (G13). • points - 12orbits of length 13; • lines - 62 orbits of length 13; • 26 line orbits consist from disjoint lines.

  14. Parallelisms ofPG(3,5) with automorphisms of order 13 Construction • Construction of spreads: • m+1 line - contains the first point, which is in none of the m spread lines; • fixed spread – add the whole line orbit (2 orbits needed); • non fixed spread – lines are from different orbits; • lexicographically ordered; • orbit leader – a fixed spread or the first in lexicographic order spread from an orbit under G13.

  15. Parallelisms ofPG(3,5) with automorphisms of order 13 Construction • Construction of parallelisms: • 7 orbit leaders ; • 2 orbits of 13 spreads; • 5 fixed spreads consisting of 2 line orbits;

  16. Parallelisms ofPG(3,5) with automorphisms of order 13 Construction • Isomorphic solutions rejection G, P, P1 – parallelismsof PG(3,5)with automorphismgroup G13, P1 = φ P  G13  P= P P= -1P P - G13 , -1 G13  N (G13) – normalizer of G13 in G G13   N (G13)  G13  -1   N (G13), P = P

  17. Parallelisms ofPG(3,5) with automorphisms of order 13 Classification results • 321 nonisomorphic parallelisms with automorphisms of order 13 only. • no regular ones among them.

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