scholarly journals Polarities and ovals in the Hughes plane

1972 ◽  
Vol 13 (2) ◽  
pp. 196-204 ◽  
Author(s):  
T. G. Room
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Proper Subset ◽  
Small Subset ◽  
Order Form ◽  
The Absolute ◽  
Hughes Plane

SummaryIn 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.

AN ALGORITHM FOR CONSTRUCTING BLOCKING SETS FOR PROJECTIVE PLANES

1992 ◽  
Vol 02 (04) ◽  
pp. 437-442
Author(s):  
RUTH SILVERMAN ◽  
ALAN H. STEIN
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Proper Subset ◽  
Blocking Set ◽  
Blocking Sets ◽  
Unique Point ◽  
Unique Line ◽  
The Family ◽  
Property B

A family of sets is said to have property B(s) if there is a set, referred to as a blocking set, whose intersection with each member of the family is a proper subset of that blocking set and contains fewer than s elements. A finite projective plane is a construction satisfying the two conditions that any two lines meet in a unique point and any two points are on a unique line. In this paper, the authors develop an algorithm of complexity O(n3) for constructing a blocking set for a projective plane of order n.

scholarly journals Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

10.37236/2582 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tamás Héger ◽  
Marcella Takáts
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
The Other ◽  
Blocking Set ◽  
Metric Dimension ◽  
Desarguesian Projective Plane ◽  
Resolving Set ◽  
Incidence Graph ◽  
Vertex Set

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

Projective planes of infinite but isolic order

1976 ◽  
Vol 41 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. C. E. Dekker
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Latin Squares ◽  
Combinatorial Theory ◽  
Ternary Ring ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Planar Ternary Ring ◽  
Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

On projective planes of type (6, m)

1980 ◽  
Vol 88 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Peter Lorimer
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes

AbstractIf a finite projective plane is of type (6, m), then m = 2 or 3.

Finite Projective Planes with Affine Subplanes

1964 ◽  
Vol 7 (4) ◽  
pp. 549-559 ◽  
Author(s):  
T. G. Ostrom ◽  
F. A. Sherk
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Desarguesian Projective Planes

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

scholarly journals On Unitary Polarities of Finite Projective Planes

1971 ◽  
Vol 23 (6) ◽  
pp. 1060-1077 ◽  
Author(s):  
William M. Kantor
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Absolute Point ◽  
Finite Projective Planes ◽  
Unitary Polarity

A unitary polarity of a finite projective plane of order q2 is a polarity θ having q3 + 1 absolute points and such that each nonabsolute line contains precisely q + 1 absolute points. Let G(θ) be the group of collineations of centralizing θ. In [15] and [16], A. Hoffer considered restrictions on G(θ) which force to be desarguesian. The present paper is a continuation of Hoffer's work. The following are our main results.THEOREM I. Let θ be a unitary polarity of a finite projective planeof order q2. Suppose that Γ is a subgroup of G(θ) transitive on the pairs x, X, with x an absolute point and X a nonabsolute line containing x. Thenis desarguesian and Γ contains PSU(3, q).

On Baer Involutions of Finite Projective Planes

1970 ◽  
Vol 22 (4) ◽  
pp. 878-880 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Finite Projective Planes ◽  
Special Cases ◽  
Collineation Groups ◽  
Baer Involution ◽  
Finite Desarguesian Projective Plane

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

scholarly journals On some hyperbolic planes from finite projective planes

2001 ◽  
Vol 25 (12) ◽  
pp. 757-762 ◽  
Author(s):  
Basri Celik
Keyword(s):  
Projective Plane ◽  
Hyperbolic Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Baer Subplane ◽  
Combinatorial Properties ◽  
Finite Projective Planes ◽  
Hyperbolic Planes

LetΠ=(P,L,I)be a finite projective plane of ordern, and letΠ′=(P′,L′,I′)be a subplane ofΠwith ordermwhich is not a Baer subplane (i.e.,n≥m2+m). Consider the substructureΠ0=(P0,L0,I0)withP0=P\{X∈P|XIl,  l∈L′},L0=L\L′whereI0stands for the restriction ofItoP0×L0. It is shown that everyΠ0is a hyperbolic plane, in the sense of Graves, ifn≥m2+m+1+m2+m+2. Also we give some combinatorial properties of the line classes inΠ0hyperbolic planes, and some relations between its points and lines.

Affine Subplanes of Finite Projective Planes

1965 ◽  
Vol 17 ◽  
pp. 977-1009 ◽  
Author(s):  
J. F. Rigby
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Finite Projective Planes

Let π be a finite projective plane of order n containing a finite projective subplane π* of order u < n. Bruck has shown (1, p. 398) that if π contains a point that does not lie on any line of π*, then n ≥ u2 + u, while if every point of π lies on a line of π* then n = u2.Let π be a finite projective plane of order n containing a finite affine subplane π0 of order m < n.

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