Subplanes of the Hughes plane of order 9

The literature of finite projective planes consists largely of general investigations, taking in as many as possible of these systems at once. However, the geometry in a specific finite plane may well be an amusing, and not entirely trivial, field of study on its own. Some papers (5, 9, 13) have in fact appeared on the geometry of the translation plane of order 9: but the Hughes plane of the same order has comparatively been neglected. The object of the present paper is to make a beginning with the study of this plane from a synthetic point of view.

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On Arcs in a Finite Projective Plane

Canadian Journal of Mathematics ◽

10.4153/cjm-1967-030-2 ◽

1967 ◽

Vol 19 ◽

pp. 376-393 ◽

Cited By ~ 15

Author(s):

G. E. Martin

Keyword(s):

Projective Plane ◽

Finite Projective Plane ◽

Projective Planes ◽

Galois Field ◽

Analytic Geometry ◽

Finite Plane ◽

Finite Projective Planes ◽

Desarguesian Planes

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments.To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.

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On the Construction of Finite Projective Planes from Homology Semibiplanes

European Journal of Combinatorics ◽

10.1016/s0195-6698(13)80044-3 ◽

1990 ◽

Vol 11 (6) ◽

pp. 589-600

Author(s):

G. Eric Moorhouse

Keyword(s):

Projective Planes ◽

Finite Projective Planes

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Generalized crystallography of diamond-like structures: II. Diamond packing in the space of a three-dimensional sphere, subconfiguration of finite projective planes, and generating clusters of diamond-like structures

Crystallography Reports ◽

10.1134/1.1509381 ◽

2002 ◽

Vol 47 (5) ◽

pp. 709-719 ◽

Cited By ~ 1

Author(s):

A. L. Talis

Keyword(s):

Three Dimensional ◽

Projective Planes ◽

Dimensional Sphere ◽

Finite Projective Planes ◽

Generalized Crystallography

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Embedding (r, 1)-designs in finite projective planes

Discrete Mathematics ◽

10.1016/0012-365x(77)90120-0 ◽

1977 ◽

Vol 19 (1) ◽

pp. 67-76 ◽

Cited By ~ 9

Author(s):

D. McCarthy ◽

S.A. Vanstone

Keyword(s):

Projective Planes ◽

Finite Projective Planes

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Pseudo-Homogeneous Coordinates for Hughes Planes

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-040-9 ◽

1996 ◽

Vol 39 (3) ◽

pp. 330-345 ◽

Cited By ~ 1

Author(s):

Peter Maier ◽

Markus Stroppel

Keyword(s):

Projective Planes ◽

Baer Subplane ◽

Homogeneous Coordinates ◽

Desargues Configuration ◽

Coset Spaces ◽

Insight Into ◽

Hughes Plane

AbstractAmong the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

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

CAUCHY ◽

10.18860/ca.v4i3.3633 ◽

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

In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system S(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.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.

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