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

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

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

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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

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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

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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

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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.

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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

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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.

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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.

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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.

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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

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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)

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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

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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

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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

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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

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q

qv

d

3571

12

12

q

q d

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

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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

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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)

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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)

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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

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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)

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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)

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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