scholarly journals Embedding Cycles in Finite Planes

10.37236/3377 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Felix Lazebnik ◽  
Keith E. Mellinger ◽  
Oscar Vega
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Finite Affine ◽  
Finite Planes

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$,  a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$,  a $k$-cycle can be embedded in any projective plane of order $q$.

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>

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).

Compactness in Topological Hjelmslev Planes

1984 ◽  
Vol 27 (4) ◽  
pp. 423-429 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Coordinate Ring ◽  
Locally Compact ◽  
Compact Projective Plane ◽  
Negative Results ◽  
Connected Projective Plane ◽  
Topological Affine

AbstractIn the theory of ordinary topological affine and projective planes it is known that (1) An affine plane is never compact (2) a locally compact ordered projective plane is compact and archimedean (3) a locally compact connected projective plane is compact and (4) a locally compact projective plane over a coordinate ring with bi-associative multiplication is compact. In this paper we re-examine these results within the theory of topological Hjelmslev Planes and observe that while (1) remains valid (2), (3) and (4) are false. At first glance these negative results seem to suggest we are working in too general a setting. However a closer examination reveals that the absence of compactness in our setting is a natural and expected feature which in no way precludes the possibility of obtaining significant results.

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 Distance in Some Finite Planes and Graphs Arising from Those Planes

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
I. Dogan ◽  
A. Akpinar
Keyword(s):  
Euclidean Plane ◽  
Projective Planes ◽  
Central Position ◽  
Finite Affine ◽  
Finite Planes ◽  
Definition Of

In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

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.

Baer Subplanes in Finite Projective and Affine Planes

1972 ◽  
Vol 24 (1) ◽  
pp. 90-97 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Baer Subplane ◽  
Affine Planes ◽  
Baer Subplanes ◽  
Finite Planes

Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘.Let π be a finite projective (respectively, an affine) plane of order n and π0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.

The Non-Existence of Finite Projective Planes of Order 10

1989 ◽  
Vol 41 (6) ◽  
pp. 1117-1123 ◽  
Author(s):  
C. W. H. Lam ◽  
L. Thiel ◽  
S. Swiercz
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Finite Projective Planes

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that1. every line contains n + 1 points,2. every point is on n + 1 lines,3. any two distinct lines intersect at exactly one point, and4. any two distinct points lie on exactly one line.It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

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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