On homotopy nilpotency of the octonian plane $\mathbb{O}P^2$

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Marek Golasi´nski
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Nilpotency Class

Let $\mathbb{O}P^2_{(p)}$ be the $p$-localization of the Cayley projective plane $\mathbb{O}P^2$ for a prime $p$ or $p=0$. We show that the homotopy nilpotency class $\textrm{nil} \Omega(\mathbb{O}P^2_{(p)})<\infty $ for $p>2$ and $\textrm{nil} \Omega (\mathbb{O}P^2_{(p)})=1$ for $p>5$ or $p=0$. The homotopy nilpotency of remaining Rosenfeld projective planes are discussed as well.

scholarly journals Maximal arcs in projective planes of order 16 and related designs

2019 ◽  
Vol 19 (3) ◽  
pp. 345-351 ◽  
Author(s):  
Mustafa Gezek ◽  
Vladimir D. Tonchev ◽  
Tim Wagner
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Maximal Arcs ◽  
Even Order

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

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.

Weakly Isotopic Planar Ternary Rings

1975 ◽  
Vol 27 (1) ◽  
pp. 32-36
Author(s):  
Frederick W. Stevenson
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Ternary Ring ◽  
Algebraic Relation ◽  
Planar Ternary Ring ◽  
Algebraic Properties

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

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 Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

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 A homomorphism theorem for projective planes

1971 ◽  
Vol 4 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Don Row
Keyword(s):  
Projective Plane ◽  
Homomorphic Image ◽  
Inverse Image ◽  
Projective Planes ◽  
Homomorphism Theorem ◽  
Central Collineations

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

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.

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