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.
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