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

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• Parallelisms of PG(3,4) with automorphisms of order 7Svetlana Topalova, Stela ZhelezovaInstitute of Mathematics and Informatics, BAS,Bulgaria

• Parallelisms of PG(3,4) with automorphisms of order 7IntroductionHistoryPG(3,4) and related BIBDs Construction of parallelisms in PG(3,4)Results

• Parallelisms of PG(3,4) with automorphisms of order 72-(v,k,) design (BIBD);V finite set of v pointsB finite collection of b blocks: k-element subsets of V D = (V, B ) 2-(v,k,) design if any 2-subset of V is in blocks of BIntroduction

• Parallelisms of PG(3,4) with automorphisms of order 7Incidence matrix A vb of a 2-(v,k,) design aij = 1 - point i in block jaij = 0 - point i not in block j r = (v-1)/(k-1) b = v.r / kIntroduction

• Parallelisms of PG(3,4) with automorphisms of order 7IntroductionIsomorphic designs exists a one-to-one correspondence between the point and block sets of both designs, which does not change the incidence.Automorphism isomorphism of the design to itself.Parallel class - partition of the point set by blocksResolution partition of the collection of blocks into parallel classesResolvability at least one resolution.

• Parallelisms of PG(3,4) with automorphisms of order 7Isomorphic 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 the design, which maps parallel classes into parallel classes.Orthogonal resolutions any two parallel classes, one from the first, and the other from the second resolution, have at most one common block.Introduction

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

• Parallelisms of PG(3,4) with automorphisms of order 7IntroductionV(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) 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; Any line in PG (d,q) has q+1 points.

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

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

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

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

• Parallelisms of PG(3,4) with automorphisms of order 7General 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.History

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

• Parallelisms of PG(3,4) with automorphisms of order 7PG(3,4) points, lines.

G group of automorphisms of PG(3,4):|G| = 1974067200Gi subgroup of order i.

G group of automorphisms of the related to PG(3,4) designs.PG(3,4) and related BIBDs

• Parallelisms of PG(3,4) with automorphisms of order 7PG(3,4) and related BIBDs Parallelisms of PG(3,4)21 spreads with 17 elements

• Parallelisms of PG(3,4) with automorphisms of order 7Cyclic subgroup of automorphisms of order 7 (G7).

Construction of parallelisms in PG(3,4)Generator of the group G7:=(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)

• Parallelisms of PG(3,4) with automorphisms of order 7Lines of PG(3,4) blocks of 2-(85,5,1) design:Construction of parallelisms in PG(3,4)

• Parallelisms of PG(3,4) with automorphisms of order 7Construction of all spreads:begin with 1 to 21 line;all spread lines from different orbits of G7.

Construction of parallelisms in PG(3,4)

• Parallelisms of PG(3,4) with automorphisms of order 7Back track search on the orbit leaders

26 028 parallelisms