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

order 13order 13Svetlana Topalova, Stela ZhelezovaInstitute of Mathematics and Informatics, BAS,Bulgaria

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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

IntroductionIntroduction

t-spreadt-spread in PG(n,q) - a set of distinct t-dimensional subspaces

which partition the point set.

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

subspaces by t-spreads.

Spread, parallelismSpread, parallelism ≡ line spread, line parallelism ≡ 1-spread, 1-

parallelism

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

IsomorphicIsomorphic parallelisms – exists an automorphism of PG(n,q)

which maps each spread of the first parallelism to a spread of

the second one.

Automorphism groupAutomorphism group of the parallelism – maps each spread of

the parallelism to a spread of the same parallelism.

TransitiveTransitive parallelism – it has an automorphism group which is

transitive on the spreads.

IntroIntrodductionuction

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

RegulusRegulus – a set RR of q+1q+1 mutually skew lines – any line

intersecting three elements of RR intersects all elements of RR.

Regular spreadRegular spread – for every three spread lines, the unique

regulus determined by them is a subset of the spread.

RegularRegular parallelismparallelism – all its spreads are regular.

IntroIntrodductionuction

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2-design2-design:

• VV – finite set of vv points

• BB – finite collection of bb blocksblocks: kk-element subsets of VV

• D = (V, BD = (V, B )) – 2-(v,k,2-(v,k,λλ)) design if any 2-subset of VV is in λλ blocks

of BB.

Parallel classParallel class – a partition of the point set by blocks.

ResolutionResolution – a partition of the collection of blocks by parallel classes.

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

IntroductionIntroduction

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

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(2PG(2nn-1,q) -1,q) – Beutelspacher, A., On parallelisms in finite projective spaces, 1974.

HistoryHistory

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

Parallelisms in PG(3,q)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.

HistoryHistory

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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.

HistoryHistory

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

The incidence of the pointspoints and t-dimensional subspacest-dimensional subspaces of

PG(n,q)PG(n,q) defines a 2-design2-design (D).

points of D

blocks of D

resolutions of D

points of PG(3,5)PG(3,5)

lines of PG(3,5)PG(3,5)

parallelisms of PG(3,5)PG(3,5)

2-(156,6,1)2-(156,6,1) design

PG(PG(33,5) and related,5) and related 2-design2-designss

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

t-dimensional subspaces

1( lines )

2(hyperplanes)

2-(v,k,) design 2-(12-(1556,6,1)6,6,1)b=806,r=31

2-(12-(1556,36,311,6),6)b=156,r=31

Parallelisms of PG(PG(33,5),5)

31 spreads with31 spreads with 2626 lineslines

PG(PG(33,5) and related,5) and related 2-design2-designss

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PG(PG(33,5),5) points, lines.

GG – group of automorphisms of PG(PG(33,,55)):

|GG| = 29 . 32 . 56 . 13 . 31

GGii – subgroup of order ii.

GG1313 – GAP – http://www.gap-system.org

GG – group of automorphisms of the related to PG(PG(33,5),5) designs.

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

1561

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q

qv

d

8061

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12

q

q d

PG(PG(33,5) and related,5) and related 2-design2-designss

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

Sylow subgroup of order 13 (GG1313).

• points - 1212 orbits of length 1313;

• lines - 6262 orbits of length 1313;

• 2626 line orbits consist from disjoint lines.

ConstructionConstruction

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

Construction of spreads:

• m+1m+1 line - contains the first point, which is in none of the mm spread

lines;

• fixed spread – add the wholewhole line orbit (2 orbits needed);

• nonnon fixed spread – lines are from differentdifferent orbits;

• lexicographically orderedordered;

• orbit leaderorbit leader – a fixed spread or the first in lexicographic order spread

from an orbit under GG1313.

ConstructionConstruction

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

Construction of parallelisms:

• 7 orbit leaders 7 orbit leaders ;

• 2 orbits of 13 spreads;

• 5 fixed spreads consisting of 2 line orbits;

ConstructionConstruction

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

Isomorphic solutions rejection

GG,

PP, , PP11 – parallelisms of PG(3,5)PG(3,5) with automorphism group GG1313

, PP11 = φφ P P

GG13 13

P P = P P PP = -1-1 P P

PP - GG13 13

, -1 -1 GG13 13

N (GN (G1313)) – normalizer of GG1313 in GG }|{)( 131

1313 GggGGgGN

ConstructionConstruction

GG13 13

N (GN (G1313))

GG13 13

-1-1 N (GN (G1313), ), P = P P = P

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

Classification resultsClassification results

321 nonisomorphic parallelisms with automorphisms of order 13 only.

no regular ones among them.