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

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

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

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

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

6. 6  2-design  2-design: V vV – finite set of v points Bb blockskVB – finite collection of b blocks: k-element subsets of V D = (V, B )2-(v,k,λ)Vλ BD = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B.  Parallel class  Parallel class – a partition of the point set by blocks.  Resolution  Resolution – a partition of the collection of blocks by parallel classes. Parallelisms of PG(3,5) with automorphisms of order 13 Introduction

7. 7 General constructions of parallelisms:  PG(n,2)  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(2 n -1,q)  PG(2 n -1,q) – Beutelspacher, A., On parallelisms in finite projective spaces, 1974. History

8. 8 Parallelisms of PG(3,5) with automorphisms of order 13 PG(3,q) Parallelisms in PG(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. History

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

10. 10 Parallelisms of PG(3,5) with automorphisms of order 13  PG(3,3)  PG(3,3) – with some group of automorphisms by Prince, 1997.  PG(3,4)  PG(3,4) – with automorphisms of orders 7 and 5 by us, 2009, 2013.  PG(3,5)  PG(3,5) – classification of cyclic parallelisms by Prince, 1998.  PG(5,2)  PG(5,2) – Stinson and Vanstone, 1986; Sarmiento, 2000 History

11. 11 Parallelisms of PG(3,5) with automorphisms of order 13 points t-dimensional subspaces PG(n,q)2-design  The incidence of the points and t-dimensional subspaces of PG(n,q) defines a 2-design (D). points of D blocks of D resolutions of D PG(3,5) points of PG(3,5) PG(3,5) lines of PG(3,5) PG(3,5) parallelisms of PG(3,5) 2-(156,6,1) 2-(156,6,1) design PG(3,5) and related 2-design s

12. 12 Parallelisms of PG(3,5) with automorphisms of order 13 t-dimensional subspaces 1 ( lines ) 2 (hyperplanes) 2-(v,k, ) design 2-(156,6,1) b=806,r=31 2-(156,31,6) b=156,r=31 PG(3,5) Parallelisms of PG(3,5) 31 spreads with 26 lines PG(3,5) and related 2-design s

13. 13 PG(3,5) PG(3,5) points, lines.  GPG(3,5)  G – group of automorphisms of PG(3,5): G |G| = 2 9. 3 2. 5 6. 13. 31 G i i G i – subgroup of order i. G 13 G 13 – GAP – http://www.gap-system.orghttp://www.gap-system.org  GPG(3,5)  G – group of automorphisms of the related to PG(3,5) designs. Parallelisms of PG(3,5) with automorphisms of order 13 PG(3,5) and related 2-design s

14. 14 Parallelisms of PG(3,5) with automorphisms of order 13 G 13  Sylow subgroup of order 13 (G 13 ). 1213points - 12 orbits of length 13; 6213lines - 62 orbits of length 13; 2626 line orbits consist from disjoint lines. Construction

15. 15 Parallelisms of PG(3,5) with automorphisms of order 13  Construction of spreads: m+1mm+1 line - contains the first point, which is in none of the m spread lines; wholefixed spread – add the whole line orbit (2 orbits needed); nondifferentnon fixed spread – lines are from different orbits; orderedlexicographically ordered; orbit leader G 13orbit leader – a fixed spread or the first in lexicographic order spread from an orbit under G 13. Construction

16. 16 Parallelisms of PG(3,5) with automorphisms of order 13  Construction of parallelisms: 7 orbit leaders7 orbit leaders ; 2 orbits of 13 spreads; 5 fixed spreads consisting of 2 line orbits; Construction

17. 17 Parallelisms of PG(3,5) with automorphisms of order 13  Isomorphic solutions rejection  G   G, P, P 1 PG(3,5)G 13 P 1 φ P P, P 1 – parallelisms of PG(3,5) with automorphism group G 13, P 1 = φ P  G 13   P   P  P  -1   P   G 13   P =   P  P =  -1   P PG 13  -1 G 13  P - G 13,  -1 G 13  N (G 13 )G 13 G N (G 13 ) – normalizer of G 13 in G Construction G 13   N (G 13 ) G 13    N (G 13 )  G 13  -1    N (G 13 ),  P = P  G 13  -1     N (G 13 ),  P = P

18. 18 Parallelisms of PG(3,5) with automorphisms of order 13 Classification results  321 nonisomorphic parallelisms with automorphisms of order 13 only.  no regular ones among them.