On Projective Planes by CAFER C¸ALIS¸KAN
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A Characterisation of Tangent Subplanes of PG(2,Q
A Characterisation of Tangent Subplanes of PG(2,q3) S.G. Barwick and Wen-Ai Jackson School of Mathematics, University of Adelaide Adelaide 5005, Australia June 23, 2018 Corresponding Author: Dr Susan Barwick, University of Adelaide, Adelaide 5005, Aus- tralia. Phone: +61 8 8303 3983, Fax: +61 8 8303 3696, email: [email protected] Keywords: Bruck-Bose representation, PG(2, q3), order q subplanes AMS code: 51E20 Abstract In [2], the authors determine the representation of order-q-subplanes and order- q-sublines of PG(2,q3) in the Bruck-Bose representation in PG(6,q). In particular, they showed that an order-q-subplane of PG(2,q3) corresponds to a certain ruled arXiv:1204.4953v1 [math.CO] 23 Apr 2012 surface in PG(6,q). In this article we show that the converse holds, namely that any ruled surface satisfying the required properties corresponds to a tangent order- q-subplane of PG(2,q3). 1 Introduction We begin with a brief introduction to 2-spreads in PG(5, q), and the Bruck-Bose repre- sentation of PG(2, q3) in PG(6, q), and introduce the notation we will use. A 2-spread of PG(5, q) is a set of q3 + 1 planes that partition PG(5, q). The following construction of a regular 2-spread of PG(5, q) will be needed. Embed PG(5, q) in PG(5, q3) and let g be a line of PG(5, q3) disjoint from PG(5, q). The Frobenius automorphism of 2 GF(q3) where x 7→ xq induces a collineation of PG(5, q3). -
Mutually Orthogonal Latin Squares and Their Generalizations
Ghent University Faculty of Sciences Department of Mathematics Mutually orthogonal latin squares and their generalizations Jordy Vanpoucke Academic year 2011-2012 Advisor: Prof. Dr. L. Storme Master thesis submitted to the Faculty of Sciences to obtain the degree of Master in Sciences, Math- ematics Contents 1 Preface 4 2 Latin squares 7 2.1 Definitions . .7 2.2 Groups and permutations . .8 2.2.1 Definitions . .8 2.2.2 Construction of different reduced latin squares . .9 2.3 General theorems and properties . 10 2.3.1 On the number of latin squares and reduced latin squares . 10 2.3.2 On the number of main classes and isotopy classes . 11 2.3.3 Completion of latin squares and critical sets . 13 3 Sudoku latin squares 16 3.1 Definitions . 16 3.2 General theorems and properties . 17 3.2.1 On the number of sudoku latin squares and inequivalent sudoku latin squares . 17 3.3 Minimal sudoku latin squares . 18 3.3.1 Unavoidable sets . 20 3.3.2 First case: a = b =2 .......................... 25 3.3.3 Second case: a = 2 and b =3 ..................... 26 3.3.4 Third case: a = 2 and b =4...................... 28 3.3.5 Fourth case: a = 3 and b =3 ..................... 29 2 4 Latin squares and projective planes 30 4.1 Projective planes . 30 4.1.1 Coordinatization of projective planes . 31 4.1.2 Planar ternary rings . 32 4.2 Orthogonal latin squares and projective planes . 33 5 MOLS and MOSLS 34 5.1 Definitions . 34 5.2 Bounds . 35 5.3 Examples of small order . -
A Characterisation of Translation Ovals in Finite Even Order Planes
A characterisation of translation ovals in finite even order planes S.G. Barwick and Wen-Ai Jackson School of Mathematics, University of Adelaide Adelaide 5005, Australia Abstract In this article we consider a set C of points in PG(4,q), q even, satisfying cer- tain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the correspond- ing Bruck-Bose plane PG(2,q2), the points corresponding to C form a translation hyperoval, and conversely. 1 Introduction In this article we first consider a non-degenerate conic in PG(2, q2), q even. We look at the corresponding point set in the Bruck-Bose representation in PG(4, q), and study its combinatorial properties (details of the Bruck-Bose representation are given in Section 2). Some properties of this set were investigated in [4]. In this article we are interested in combinatorial properties relating to planes of PG(4, q). We consider a set of points in PG(4, q) satisfying certain of these combinatorial properties and find that the points 2 arXiv:1305.6673v1 [math.CO] 29 May 2013 correspond to a translation oval in the Bruck-Bose plane PG(2, q ). In [3], the case when q is odd is considered, and we show that given a set of points in PG(4, q) satisfying the following combinatorial properties, we can reconstruct the conic in PG(2, q2). We use the following terminology in PG(4, q): if the hyperplane at infinity is denoted Σ∞, then we call the points of PG(4, q) \ Σ∞ affine points. -
Projective Planarity of Matroids of 3-Nets and Biased Graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 76(2) (2020), Pages 299–338 Projective planarity of matroids of 3-nets and biased graphs Rigoberto Florez´ ∗ Deptartment of Mathematical Sciences, The Citadel Charleston, South Carolina 29409 U.S.A. [email protected] Thomas Zaslavsky† Department of Mathematical Sciences, Binghamton University Binghamton, New York 13902-6000 U.S.A. [email protected] Abstract A biased graph is a graph with a class of selected circles (“cycles”, “cir- cuits”), called “balanced”, such that no theta subgraph contains exactly two balanced circles. A 3-node biased graph is equivalent to an abstract partial 3-net. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle is equivalent to a 3-net, which, in turn, is equivalent to an isostrophe class of quasigroups. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
148. Symplectic Translation Planes by Antoni
Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 2 (2003), pp. 101 - 148. Symplectic translation planes by Antonio Maschietti Abstract1. A great deal of important work on symplectic translation planes has occurred in the last two decades, especially on those of even order, because of their link with non-linear codes. This link, which is the central theme of this paper, is based upon classical groups, particularly symplectic and orthogonal groups. Therefore I have included an Appendix, where standard notation and basic results are recalled. 1. Translation planes In this section we give an introductory account of translation planes. Compre- hensive textbooks are for example [17] and [3]. 1.1. Notation. We will use linear algebra to construct interesting geometrical structures from vector spaces, with special regard to vector spaces over finite fields. Any finite field has prime power order and for any prime power q there is, up to isomorphisms, a unique finite field of order q. This unique field is commonly de- noted by GF (q)(Galois Field); but also other symbols are usual, such as Fq.If q = pn with p a prime, the additive structure of F is that of an n dimensional q − vector space over Fp, which is the field of integers modulo p. The multiplicative group of Fq, denoted by Fq⇤, is cyclic. Finally, the automorphism group of the field Fq is cyclic of order n. Let A and B be sets. If f : A B is a map (or function or else mapping), then the image of x A will be denoted! by f(x)(functional notation). -
Matroids You Have Known
26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction. -
The Turan Number of the Fano Plane
Combinatorica (5) (2005) 561–574 COMBINATORICA 25 Bolyai Society – Springer-Verlag THE TURAN´ NUMBER OF THE FANO PLANE PETER KEEVASH, BENNY SUDAKOV* Received September 17, 2002 Let PG2(2) be the Fano plane, i.e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3-uniform hypergraph on n vertices not containing a Fano plane is n n/2 n/2 ex n, P G (2) = − − . 2 3 3 3 Moreover, the only extremal configuration can be obtained by partitioning an n-element set into two almost equal parts, and taking all the triples that intersect both of them. This extends an earlier result of de Caen and F¨uredi, and proves an old conjecture of V. S´os. In addition, we also prove a stability result for the Fano plane, which says that a 3-uniform hypergraph with density close to 3/4 and no Fano plane is approximately 2-colorable. 1. Introduction Given an r-uniform hypergraph F,theTur´an number of F is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. We denote this number by ex(n,F). Determining these numbers is one of the central problems in Extremal Combinatorics, and it is well understood for ordinary graphs (the case r =2). It is completely solved for many instances, including all complete graphs. Moreover, asymptotic results are known for all non-bipartite graphs. -
Finite Projective Geometry 2Nd Year Group Project
Finite Projective Geometry 2nd year group project. B. Doyle, B. Voce, W.C Lim, C.H Lo Mathematics Department - Imperial College London Supervisor: Ambrus Pal´ June 7, 2015 Abstract The Fano plane has a strong claim on being the simplest symmetrical object with inbuilt mathematical structure in the universe. This is due to the fact that it is the smallest possible projective plane; a set of points with a subsets of lines satisfying just three axioms. We will begin by developing some theory direct from the axioms and uncovering some of the hidden (and not so hidden) symmetries of the Fano plane. Alternatively, some projective planes can be derived from vector space theory and we shall also explore this and the associated linear maps on these spaces. Finally, with the help of some theory of quadratic forms we will give a proof of the surprising Bruck-Ryser theorem, which shows that if a projective plane has order n congruent to 1 or 2 mod 4, then n is the sum of two squares. Thus we will have demonstrated fascinating links between pure mathematical disciplines by incorporating the use of linear algebra, group the- ory and number theory to explain the geometric world of projective planes. 1 Contents 1 Introduction 3 2 Basic Defintions and results 4 3 The Fano Plane 7 3.1 Isomorphism and Automorphism . 8 3.2 Ovals . 10 4 Projective Geometry with fields 12 4.1 Constructing Projective Planes from fields . 12 4.2 Order of Projective Planes over fields . 14 5 Bruck-Ryser 17 A Appendix - Rings and Fields 22 2 1 Introduction Projective planes are geometrical objects that consist of a set of elements called points and sub- sets of these elements called lines constructed following three basic axioms which give the re- sulting object a remarkable level of symmetry. -
LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS of QUASIFIELDS 3 Group T (Π) of Translations
LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS OF QUASIFIELDS GABOR´ P. NAGY Abstract. For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of fi- nite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the iso- morphism of the corresponding translation planes. 1. Introduction A translation plane is often represented by an algebraic structure called a quasifield. Many properties of the translation plane can be most easily understood by looking at the appropriate quasifield. However, isomorphic translation planes can be represented by nonisomorphic quasifields. Further- more, the collineations do not always have a nice representation in terms of operations in the quasifields. Let p be a prime number and (Q, +, ·) be a quasifield of finite order pn. Fn We identify (Q, +) with the vector group ( p , +). With respect to the multiplication, the set Q∗ of nonzero elements of Q form a loop. The right multiplication maps of Q are the maps Ra : Q → Q, xRa = x · a, where Fn a, x ∈ Q. By the right distributive law, Ra is a linear map of Q = p . Clearly, R0 is the zero map. If a 6= 0 then Ra ∈ GL(n,p). In geometric context, the set of right translations are also called the slope set or the spread set of the quasifield Q, cf. [6, Chapter 5]. The right multiplication group RMlt(Q) of the quasifield Q is the linear group generated by the nonzero right multiplication maps. -
Finite Semifields and Nonsingular Tensors
FINITE SEMIFIELDS AND NONSINGULAR TENSORS MICHEL LAVRAUW Abstract. In this article, we give an overview of the classification results in the theory of finite semifields1and elaborate on the approach using nonsingular tensors based on Liebler [52]. 1. Introduction and classification results of finite semifields 1.1. Definition, examples and first classification results. Finite semifields are a generalisation of finite fields (where associativity of multiplication is not assumed) and the study of finite semifields originated as a classical part of algebra in the work of L. E. Dickson and A. A. Albert at the start of the 20th century. Remark 1.1. The name semifield was introduced by Knuth in his dissertation ([41]). In the literature before that, the algebraic structure, satisfying (S1)-(S4), was called a distributive quasifield, a division ring or a division algebra. Since the 1970's the use of the name semifields has become the standard. Due to the Dickson-Wedderburn Theorem which says that each finite skew field is a field (see [36, Section 2] for some historical remarks), finite semifields are in some sense the algebraic structures closest to finite fields. It is therefore not surprising that Dickson took up the study of finite semifields shortly after the classification of finite fields at the end of the 19th century (announced by E. H. Moore on the International Congress for Mathematicians in Chicago in 1893). Remark 1.2. In the remainder of this paper we only consider finite semifields (unless stated otherwise) and finiteness will often be assumed implicitly. In the infinite case, the octonions (see e.g. -
Pedal Sets of Unitals in Projective Planes of Order 9 and 16
SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (20) (2011), 255{264 PEDAL SETS OF UNITALS IN PROJECTIVE PLANES OF ORDER 9 AND 16 VEDRAN KRCADINACˇ AND KSENIJA SMOLJAK Dedicated to the memory of Professor Svetozar Kurepa Abstract. Let U be a unital in a projective plane P. Given a point P 2 P n U, the points F 2 U such that FP is tangent to U are the feet of P , and the set of all such points is the pedal set of P . In this paper we study pedal sets of unitals embedded in projective planes of order 9 and 16. 1. Introduction A finite projective plane of order q can be defined as a block design with parameters (q2 + q + 1; q + 1; 1); that is, a finite incidence structure with q2 + q + 1 points, q + 1 points on every line, and a unique line through every pair of points. Similarly, a unital of order q is a (q3 +1; q +1; 1) block design. For more on finite projective planes we refer to the book [10], and for unitals to [2]. The most interesting unitals are the ones embedded in a projective plane P of order q2. In this setting a unital U is a set of q3 + 1 points of P with the property that every line of P meets U either in one point, or in q + 1 points. In the first case the line is tangent to U, and in the second case it is secant. The set of all tangents is a unital in the dual plane P∗, called the dual unital and denoted by U ∗.