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

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)

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

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

  2. Parallelisms ofPG(3,4) with automorphisms of order 7 • Introduction • History • PG(3,4) and related BIBDs • Construction of parallelisms in PG(3,4) • Results

  3. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction 2-(v,k,λ) design (BIBD); • V– finite set of vpoints • B – finite collection of bblocks: k-element subsets of V • D = (V, B) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B

  4. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • Incidence matrix A v×bof a 2-(v,k,λ) design aij = 1 - point i in block j aij = 0 - point i not in block j r = λ(v-1)/(k-1) b = v.r / k

  5. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • Isomorphicdesigns – 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 blocks • Resolution – partition of the collection of blocks into parallel classes • Resolvability – at least one resolution.

  6. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • Isomorphic 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.

  7. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction A finiteprojective 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.

  8. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • V(d+1,F) – vector space of dimension d+1over the finite field F (the number of elements of Fis 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.

  9. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • 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.

  10. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • 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-parallelism

  11. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • The incidence of the pointsand t-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)

  12. Parallelisms ofPG(3,4) with automorphisms of order 7 Introduction • Transitiveparallelism – it has an automorphism group which acts transitively on the spreads. • Cyclicparallelism – there is an automorphism of order the number of spreads which permutes them cyclically.

  13. Parallelisms ofPG(3,4) with automorphisms of order 7 History General constructions of parallelisms: • Constructions of parallelismsinPG(2n-1,q), Beutelspacher, 1974. • Transitive parallelisms inPG(3,q) – Denniston, 1972. • Orthogonal parallelisms – Fuji-HarainPG(3,q), 1986.

  14. Parallelisms ofPG(3,4) with automorphisms of order 7 History ParallelismsinPG(3,q): • PG(3,2)– all are classified. • PG(3,3)– with some group of automorphisms byPrince, 1997. • PG(3,4) – only examples by general constructions. • PG(3,5) – classification of cyclic parallelisms byPrince, 1998.

  15. Parallelisms ofPG(3,4) with automorphisms of order 7 PG(3,4) and relatedBIBDs PG(3,4) points, lines. • G – group of automorphisms of PG(3,4): |G| =1974067200 Gi – subgroup oforderi. • G – group of automorphisms of the related toPG(3,4)designs.

  16. Parallelisms ofPG(3,4) with automorphisms of order 7 PG(3,4) and relatedBIBDs Parallelisms ofPG(3,4) 21 spreads with 17 elements

  17. Parallelisms ofPG(3,4) with automorphisms of order 7 Construction of parallelisms in PG(3,4) • Cyclic subgroup of automorphisms of order 7 (G7). Generator of the groupG7: 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)

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

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

  20. Parallelisms ofPG(3,4) with automorphisms of order 7 Construction of parallelisms in PG(3,4) • Back track search on the orbit leaders 26 028 parallelisms spread (parallel class) – orbit leader

  21. Parallelisms ofPG(3,4) with automorphisms of order 7 Construction of parallelisms in PG(3,4)  G, P1 – parallelism of PG(3,4), automorphismgroup GP1,   GP1 P2 = φ P1 – parallelism of PG(3,4), automorphismgroup GP2,  GP2  P1 =   P1  =   -1 GP2= GP1-1 N (G7) – normalizer of G7 in G | N (G7) | = 378G54 = N (G7) \ G7

  22. Parallelisms ofPG(3,4) with automorphisms of order 7 R e s u l t s • 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.

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