Parallelisms of PG(3,4) Parallelisms of PG(3,4) with automorphisms of with automorphisms of

order 7order 7Svetlana Topalova, Stela ZhelezovaInstitute of Mathematics and Informatics, BAS,Bulgaria

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

IntroductionIntroduction

2-(v,k,λ) design (BIBD); VV – finite set of v points BB – finite collection of bb blocksblocks: kk-element subsets

of VV D = (V, BD = (V, B )) – 2-(v,k,λ) design if any 2-subset of VV

is in λλ blocks of BB

Parallelisms of PG(3,4) with automorphisms of order 7

4

IntroductionIntroduction Incidence matrix A v×b of a 2-(v,k,λ) design

aaijij = 1 = 1 - point i in block jj

aaijij = 0 = 0 - point i not in block jj

r = r = λλ(v-1)/(k-1)(v-1)/(k-1)

b = v.r / kb = v.r / k

Parallelisms of PG(3,4) with automorphisms of order 7

5

Parallelisms of PG(3,4) with automorphisms of order 7

IntroductionIntroduction

IsomorphicIsomorphic designsdesigns – exists a one-to-one correspondence between the point and block sets of both designs, which does not change the incidence.

AutomorphismAutomorphism – isomorphism of the design to itself.

Parallel classParallel class - partition of the point set by blocks

ResolutionResolution – partition of the collection of blocks into parallel classes

ResolvabilityResolvability – at least one resolution.

6

Parallelisms of PG(3,4) with automorphisms of order 7

Isomorphic resolutionsIsomorphic 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 resolutionAutomorphism of a resolution - automorphism of the design, which maps parallel classes into parallel classes.

Orthogonal resolutionsOrthogonal resolutions – any two parallel classes, one from the first, and the other from the second resolution, have at most one common block.

IntroductionIntroduction

7

Parallelisms of PG(3,4) with automorphisms of order 7

IntroductionIntroduction

A finitefinite projective spaceprojective space is a finite incidence structure (a finite

set of pointsset of points, a finite set of linesset 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

Parallelisms of PG(3,4) with automorphisms of order 7

IntroductionIntroduction

V(d+1,F)V(d+1,F) – vector space of dimension d+1d+1 over the finite field

FF (the number of elements of FF is qq);

PG(d,q)PG(d,q) – projective space of dimension dd and order qq has as its points the 1-dimensional subspaces of V, and as its lines the 2-dimensional subspaces of V;

Any line in PG (d,q)PG (d,q) has q+1q+1 points.

9

Parallelisms of PG(3,4) with automorphisms of order 7

IntroductionIntroduction

An automorphismautomorphism of PG(d,q)PG(d,q) is a bijective map on the point set

that preserves collinearity, i.e. maps the lines into lines.

A spreadspread in PG(d,q) - a set of lines which partition the point set.

A t-spreadt-spread in PG(d,q) - a set of t-dimensional subspaces which

partition the point set.

10

Parallelisms of PG(3,4) with automorphisms of order 7

A parallelismparallelism in PG(d,q) – a partition of the set of lines by

spreads.

A t-parallelismt-parallelism in PG(d,q) – a partition of the set of t-dimensional

subspaces by t-spreads.

parallelism = 1-parallelism

IntroductionIntroduction

11

Parallelisms of PG(3,4) with automorphisms of order 7

IntroductionIntroduction

The incidence of the pointspoints and t-dimensional subspacest-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

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.

IntroductionIntroduction

13

Parallelisms of PG(3,4) with automorphisms of order 7

General constructions of parallelisms:

Constructions of parallelisms in PG(2n-1,q), Beutelspacher, 1974.

Transitive parallelisms in PG(3,q) – Denniston, 1972.

Orthogonal parallelisms – Fuji-Hara in PG(3,q), 1986.

HistoryHistory

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.

HistoryHistory

15

PG(PG(33,,44)) points, lines.

GG – group of automorphisms of PG(3,4):

|G| = 1974067200

Gi – subgroup of order i.

GG – group of automorphisms of the related to PG(3,4) designs.

Parallelisms of PG(3,4) with automorphisms of order 7

851

11

q

qv

d

3571

12

12

q

q d

PG(PG(33,,44) and related) and related BIBDsBIBDs

16

Parallelisms of PG(3,4) with automorphisms of order 7

PG(PG(33,,44) and related) and related BIBDsBIBDs

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

Parallelisms of PG(3,4) with automorphisms of order 7

Cyclic subgroup of automorphisms of order 7 (G7).

Construction of parallelisms in PG(3,4)Construction of parallelisms in PG(3,4)

Generator of the group G7:

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

1 2 3 4 5 6 7 … 20 21 22 … 355 356 357

1 1 1 1 1 1 1 … 1 1 2 … 21 21 21

2 6 10 14 18 22 26 … 78 82 6 … 35 36 37

3 7 11 15 19 23 27 ... 79 83 10 … 40 39 38

4 8 12 16 20 24 28 … 80 84 14 … 62 65 64

5 9 13 17 21 25 29 … 81 85 18 … 77 74 75

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)Construction of parallelisms in PG(3,4)

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 G7.

Construction of parallelisms in PG(3,4)Construction of parallelisms in PG(3,4)

1 102 122 142 162 172 191 212 217 241 261 269 280 307 320 331 343

1 102 122 142 162 172 195 199 225 235 256 276 280 301 313 341 351

. . . . . . . . . . . . . . . . .

2 26 50 74 98 172 191 212 217 241 261 269 280 307 320 331 343

. . . . . . . . . . . . . . . . .

21 37 53 69 85 112 118 140 189 209 229 243 248 282 306 335 343

21 37 53 69 85 112 122 136 189 209 226 230 252 293 307 335 343

20

Parallelisms of PG(3,4) with automorphisms of order 7

Back track search on the orbit leadersorbit leaders

26 028 parallelisms

1 102 122 142 162 … 320 331 343

3 29 54 73 100 … 312 336 356

4 39 56 67 92 … 315 340 351

spread (parallel class) – orbit leader

Construction of parallelisms in PG(3,4)Construction of parallelisms in PG(3,4)

21

Parallelisms of PG(3,4) with automorphisms of order 7

G,

P1 – parallelism of PG(3,4), automorphism group GP1 , GP1

P2 = φ P1 – parallelism of PG(3,4), automorphism group GP2 , GP2

P1 = P1 = -1 GP2 = GP1

-1

N (G7) – normalizer of G7 in G

| N (G7) | = 378 G54 = N (G7) \ G7

}|{)( 71

77 GggGGgGN

Construction of parallelisms in PG(3,4)Construction of parallelisms in PG(3,4)

22

Parallelisms of PG(3,4) with automorphisms of order 7

All 26 028 parallelisms – orbits of length 54 under

G54 = N (G7) \ G7

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 sR e s u l t s