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

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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

    spread (parallel class) orbit leaderConstruction of parallelisms in PG(3,4)

    11021221421623203313433295473100312336356439566792315340351

  • Parallelisms of PG(3,4) with automorphisms of order 7 G,P1 parallelism of PG(3,4), automorphism group GP1 , GP1P2 = 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) \ G7Construction of parallelisms in PG(3,4)

  • Parallelisms of PG(3,4) with automorphisms of order 7All 26 028 parallelisms orbits of length 54 under G54 = N (G7) \ G7482 non isomorphic parallelisms of PG(3,4) with automorphisms of order 7.All constructed parallelisms have full group of automorphisms of order 7No pairs of orthogonal parallelisms among them.R e s u l t s

    This is joint work with Svetlana Topalova. We are both from the Institute of Matematics and Informatics . We construct all non isomorphic parallelisms of PG(3,4) with automorphisms of order 7.Let V={ Pi } vi=1 be a finite set of points, and B ={ B_j } bj=1- a finite collection of k-element subsets of V, called blocks. We say that D=(V, B) is a design with parameters 2-(v,k,) or balanced incomplete block design (BIBD), if any 2-subset of V is contained in exactly blocks of B.The incidence matrix A with v rows and b columns, where Aij = 1 if the point i is in a block j and 0 otherwise determines the design. It has r ones in each row and k ones in each column. The relations among the design parameters are presented here.This is the incidence matrix of a design on 9 points, block size 3 and lambda 3. The design has 36 blocks each incident with 3 points, so the incidence matrix is with 9 rows, 36 columns and has 12 ones in each row and 3 ones in each column.Two designs are isomorphic if there exists a one-to-one correspondence between the point and block sets of the first design and respectively, the point and block sets of the second design, and if this one-to-one correspondence does not change the incidence. An automorphism of the design is an isomorphism into itself.(permutation of the points that transforms the blocks into blocks)A parallel class is a partition of the point set by blocks. A resolution of the design is a partition of the collection of blocks by parallel classes. One of the most important properties of a design is its resolvability. The design is resolvable if it has at least one resolution. Two resolutions are isomorphic if there exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one.An automorphism of a resolution is an automorphism of the design, which maps parallel classes into parallel classes.Two resolutions of one and the same design are mutually orthogonal if any two parallel classes, one from the first, and the other from the second resolution, have at most one common block.Orthogonal resolutions may or may not be isomorphic to each other.A finite projective space is a finite incidence structure such that the following axioms hold:Let V be a vector space of dimension d+1 over the finite field F of q elements.We denote by PG(d,q) the projective space of dimension d and order q, which has as its points the 1-dimensional subspaces and as its lines the 2-dimensional subspaces of V. A spread in the projective space of dimension d and order q is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. T-spreads and t-parallelisms are defined in a similar way. If we say parallelism, we mean a 1-parallelism, namely line parallelism.An automorphism of the projective space is a bijective map on the point set that preserves collinearity, i.e. maps lines into lines, and thus t-dimensional subspaces into t-dimensional subspaces.A spread in the projective space of dimension d and order q is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. T-spreads and t-parallel