A Class of Non-Desarguesian Projective Planes

1957 ◽  
Vol 9 ◽  
pp. 378-388 ◽  
Author(s):  
D. R. Hughes
Keyword(s):  
Positive Integer ◽  
Projective Plane ◽  
Collineation Group ◽  
Difference Sets ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Desarguesian Plane ◽  
Desarguesian Projective Planes

In (7), Veblen and Wedclerburn gave an example of a non-Desarguesian projective plane of order 9; we shall show that this plane is self-dual and can be characterized by a collineation group of order 78, somewhat like the planes associated with difference sets. Furthermore, the technique used in (7) will be generalized and we will construct a new non-Desarguesian plane of order p2n for every positive integer n and every odd prime p.

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 Finite groups as automorphism groups of orthocomplemented projective planes

1978 ◽  
Vol 25 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Richard J. Greechie
Keyword(s):  
Finite Group ◽  
Projective Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Absolute Point

AbstractA construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

scholarly journals Blocking and Double Blocking Sets in Finite Planes

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde
Keyword(s):  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Desarguesian Plane ◽  
Baer Subplanes ◽  
Desarguesian Projective Planes ◽  
Finite Planes ◽  
Value Sets ◽  
Minimal Blocking ◽  
Desarguesian Affine Plane

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

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).

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 Havliček–Tietze Configurations in Various Projective Planes

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Jan Jakóbowski ◽  
Danuta Kacperek
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane

AbstractA. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havlicek-Tietze configuration (shortly H-T) in the desarguesian projective plane is equivalent to existence in the associated field, a root of polynomial x

scholarly journals Generalizing the Algebra of Throws to Rank-3 Matroids

2021 ◽  
Author(s):  
◽  
Jasmine Hall
Keyword(s):  
Projective Plane ◽  
Binary Operation ◽  
Group Structure ◽  
Projective Planes ◽  
General Point ◽  
Desarguesian Projective Planes ◽  
Line Configuration ◽  
Forbidden Configurations ◽  
Algebraic Operations ◽  
Point Line

<p>The algebra of throws is a geometric construction which reveals the underlying algebraic operations of addition and multiplication in a projective plane. In Desarguesian projective planes, the algebra of throws is a well-defined, commutative and associative binary operation. However, when we consider an analogous operation in a more general point-line configuration that comes from rank-3 matroids, none of these properties are guaranteed. We construct lists of forbidden configurations which give polynomial time checks for certain properties. Using these forbidden configurations, we can check whether a configuration has a group structure under this analogous operation. We look at the properties of configurations with such a group structure, and discuss their connection to the jointless Dowling geometries.</p>

scholarly journals Generalizing the Algebra of Throws to Rank-3 Matroids

2021 ◽  
Author(s):  
◽  
Jasmine Hall
Keyword(s):  
Projective Plane ◽  
Binary Operation ◽  
Group Structure ◽  
Projective Planes ◽  
General Point ◽  
Desarguesian Projective Planes ◽  
Line Configuration ◽  
Forbidden Configurations ◽  
Algebraic Operations ◽  
Point Line

<p>The algebra of throws is a geometric construction which reveals the underlying algebraic operations of addition and multiplication in a projective plane. In Desarguesian projective planes, the algebra of throws is a well-defined, commutative and associative binary operation. However, when we consider an analogous operation in a more general point-line configuration that comes from rank-3 matroids, none of these properties are guaranteed. We construct lists of forbidden configurations which give polynomial time checks for certain properties. Using these forbidden configurations, we can check whether a configuration has a group structure under this analogous operation. We look at the properties of configurations with such a group structure, and discuss their connection to the jointless Dowling geometries.</p>

scholarly journals On rank 4 projective planes

1981 ◽  
Vol 4 (2) ◽  
pp. 305-319
Author(s):  
O. Bachmann
Keyword(s):  
Projective Plane ◽  
Collineation Group ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Line Pair ◽  
Rank 2 ◽  
Point Line

Let a finite projective plane be called rankmplane if it admits a collineation groupGof rankm, let it be called strong rankmplane if moreoverGP=G1for some point-line pair(P,1). It is well known that every rank 2 plane is desarguesian (Theorem of Ostrom and Wagner). It is conjectured that the only rank 3 plane is the plane of order 2. By [1] and [7] the only strong rank 3 plane is the plane of order 2. In this paper it is proved that no strong rank 4 plane exists.

scholarly journals Daisy structure in Desarguesian projective planes

2003 ◽  
Vol 74 (2) ◽  
pp. 145-154 ◽  
Author(s):  
M. Gabriela Araujo Pardo
Keyword(s):  
Lower Bound ◽  
Projective Plane ◽  
Block Design ◽  
Projective Planes ◽  
Special Structure ◽  
Blocking Set ◽  
Desarguesian Projective Planes

AbstractWe distribute the points and lines ofPG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic toPG(n, 2).We show a blocking set of 3qpoints inPG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.

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