1. Parallelisms of PG(3,4) with automorphisms of order 7 Svetlana Topalova, Stela Zhelezova Institute of Mathematics and Informatics, BAS,Bulgaria

2. 2 Parallelisms of PG(3,4) with automorphisms of order 7  Introduction  History  PG(3,4) and related BIBDs  Construction of parallelisms in PG(3,4)  Results

3. 3 Introduction 2-(v,k,λ) design (BIBD);  V  V – finite set of v points  Bb blocksk V  B – finite collection of b blocks: k-element subsets of V  D = (V, B )V λB  D = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B Parallelisms of PG(3,4) with automorphisms of order 7

4. 4 Introduction  Incidence matrix A v×b of a 2-(v,k,λ) design a ij = 1j a ij = 1 - point i in block j a ij = 0j a ij = 0 - point i not in block j r = λ(v-1)/(k-1) r = λ(v-1)/(k-1) b = v.r / k Parallelisms of PG(3,4) with automorphisms of order 7

5. 5 Introduction  Isomorphic designs  Isomorphic designs – exists a one-to-one correspondence between the point and block sets of both designs, which does not change the incidence.  Automorphism  Automorphism – isomorphism of the design to itself.  Parallel class  Parallel class - partition of the point set by blocks  Resolution  Resolution – partition of the collection of blocks into parallel classes  Resolvability  Resolvability – at least one resolution.

6. 6 Parallelisms of PG(3,4) with automorphisms of order 7  Isomorphic resolutions  Isomorphic resolutions - exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one.  Automorphism of a resolution  Automorphism of a resolution - automorphism of the design, which maps parallel classes into parallel classes.  Orthogonal resolutions  Orthogonal resolutions – any two parallel classes, one from the first, and the other from the second resolution, have at most one common block. Introduction

7. 7 Parallelisms of PG(3,4) with automorphisms of order 7 Introduction finite projective space set of pointsset of lines A finite projective space is a finite incidence structure (a finite set of points, a finite set of lines, and an incidence relation between them) such that: any two distinct points are on exactly one line; let A, B, C, D be four distinct points of which no three are collinear. If the lines AB and CD intersect each other, then the lines AD and BC also intersect each other; any line has at least 3 points.

8. 8 Parallelisms of PG(3,4) with automorphisms of order 7 Introduction  V(d+1,F)d+1 FFq  V(d+1,F) – vector space of dimension d+1 over the finite field F (the number of elements of F is q); PG(d,q)dq  PG(d,q) – projective space of dimension d and order q has as its points the 1-dimensional subspaces of V, and as its lines the 2-dimensional subspaces of V; PG (d,q)q+1  Any line in PG (d,q) has q+1 points.

9. 9 Parallelisms of PG(3,4) with automorphisms of order 7 Introduction automorphismPG(d,q)  An automorphism of PG(d,q) is a bijective map on the point set that preserves collinearity, i.e. maps the lines into lines.  spread  A spread in PG(d,q) - a set of lines which partition the point set. t-spread  A t-spread in PG(d,q) - a set of t-dimensional subspaces which partition the point set.

10. 10 Parallelisms of PG(3,4) with automorphisms of order 7 parallelism  A parallelism in PG(d,q) – a partition of the set of lines by spreads. t-parallelism  A t-parallelism in PG(d,q) – a partition of the set of t-dimensional subspaces by t-spreads.  parallelism = 1-parallelism Introduction

11. 11 Parallelisms of PG(3,4) with automorphisms of order 7 Introduction points t-dimensional subspaces  The incidence of the points and t-dimensional subspaces of PG(d,q) defines a BIBD (D). points of D blocks of D resolutions of D points of PG(d,q) t-dimentional subspaces of PG(d,q) t-parallelisms of PG(d,q)

12. 12 Parallelisms of PG(3,4) with automorphisms of order 7  Transitive parallelism – it has an automorphism group which acts transitively on the spreads.  Cyclic parallelism – there is an automorphism of order the number of spreads which permutes them cyclically. Introduction

13. 13 Parallelisms of PG(3,4) with automorphisms of order 7 General constructions of parallelisms:  Constructions of parallelisms in PG(2 n -1,q), Beutelspacher, 1974.  Transitive parallelisms in PG(3,q) – Denniston, 1972.  Orthogonal parallelisms – Fuji-Hara in PG(3,q), 1986. History

14. 14 Parallelisms of PG(3,4) with automorphisms of order 7 Parallelisms in PG(3,q):  PG(3,2) – all are classified.  PG(3,3) – with some group of automorphisms by Prince, 1997.  PG(3,4) – only examples by general constructions.  PG(3,5) – classification of cyclic parallelisms by Prince, 1998. History

15. 15 PG(3,4) PG(3,4) points, lines. G G – group of automorphisms of PG(3,4): |G| = 1974067200 G i – subgroup of order i. G G – group of automorphisms of the related to PG(3,4) designs. Parallelisms of PG(3,4) with automorphisms of order 7 PG(3,4) and related BIBDs

16. 16 Parallelisms of PG(3,4) with automorphisms of order 7 PG(3,4) and related BIBDs t-dimentional subspaces 1 ( lines ) 2 (hyperplanes) 2-(v,k, ) design 2-(85,5,1) b=357,r=21 2-(85,21,5) b=85,r=21 Parallelisms of PG(3,4) 21 spreads with 17 elements

17. 17 Parallelisms of PG(3,4) with automorphisms of order 7 Cyclic subgroup of automorphisms of order 7 (G 7 ). Construction of parallelisms in PG(3,4) Generator of the group G 7 : α=(1,30,23,31,5,2,22)(3,26,24,33,37,27,29)(4,34,25,32,28,35,36)(6)(7,46,39,47,15,14,38) (8,78,71,79,20,18,70)(9,62,55,63,13,10,54)(11,74,40,81,69,59,45)(12,50,73,48,60,67,84) (16,58,72,65,53,43,77)(17,82,57,80,44,51,68)(19,66,41,64,76,83,52)(21,42,56,49,85,75,61)

18. 18 1234567…202122…355356357 1111111…112…21 261014182226…78826…353637 371115192327...798310…403938 481216202428…808414…626564 591317212529…818518…777475 Parallelisms of PG(3,4) with automorphisms of order 7  Lines of PG(3,4) ≡ blocks of 2-(85,5,1) design: Construction of parallelisms in PG(3,4)

19. 19 Parallelisms of PG(3,4) with automorphisms of order 7  Construction of all spreads: begin with 1 to 21 line; all spread lines from different orbits of G 7. Construction of parallelisms in PG(3,4) 1102122142162172191212217241261269280307320331343 1102122142162172195199225235256276280301313341351................. 226507498172191212217241261269280307320331343................. 2137536985112118140189209229243248282306335343 2137536985112122136189209226230252293307335343

20. 20 Parallelisms of PG(3,4) with automorphisms of order 7 orbit leaders Back track search on the orbit leaders 26 028 parallelisms 1102122142162…320331343 3295473100…312336356 439566792…315340351 spread (parallel class) – orbit leader Construction of parallelisms in PG(3,4)

21. 21 Parallelisms of PG(3,4) with automorphisms of order 7   G, P 1 – parallelism of PG(3,4), automorphism group G P 1,   G P 1 P 2 = φ P 1 – parallelism of PG(3,4), automorphism group G P 2,   G P 2   P 1 =   P 1  =    -1 G P 2 =  G P 1  -1 N (G 7 ) – normalizer of G 7 in G | N (G 7 ) | = 378 G 54 = N (G 7 ) \ G 7 Construction of parallelisms in PG(3,4)

22. 22 Parallelisms of PG(3,4) with automorphisms of order 7  All 26 028 parallelisms – orbits of length 54 under G 54 = N (G 7 ) \ G 7  482 non isomorphic parallelisms of PG(3,4) with automorphisms of order 7.  All constructed parallelisms have full group of automorphisms of order 7  No pairs of orthogonal parallelisms among them. R e s u l t s