Cyclic Affine Planes

1952 ◽  
Vol 4 ◽  
pp. 295-301 ◽  
Author(s):  
A. J. Hoffman
Keyword(s):  
Cyclic Group ◽  
Infinite Number ◽  
Affine Plane ◽  
Principal Result ◽  
Affine Planes ◽  
Present Discussion ◽  
Desarguesian Affine Plane

Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes

1964 ◽  
Vol 16 ◽  
pp. 443-472 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Positive Integer ◽  
Collineation Group ◽  
Affine Plane ◽  
Affine Planes ◽  
Collineation Groups ◽  
Point Line ◽  
Explicit Determination ◽  
Desarguesian Affine Plane

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).

scholarly journals On the Non-Existence of Certain Hyperovals in Dual André Planes of Order $2^{2h}$

10.37236/912 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Angela Aguglia ◽  
Luca Giuzzi
Keyword(s):  
Affine Plane ◽  
Dimension 2 ◽  
Desarguesian Affine Plane

No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual André plane of order $2^{2h}$ and dimension $2$ over its kernel.

Arcs of Parabolic Order Four

1964 ◽  
Vol 16 ◽  
pp. 321-338 ◽  
Author(s):  
N. D. Lane
Keyword(s):  
Affine Plane ◽  
Cyclic Order ◽  
Proved Theorem ◽  
Present Discussion ◽  
The Real ◽  
End Point ◽  
Real Affine ◽  
Order Four ◽  
Strong Differentiability

This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.

scholarly journals The modal logic of affine planes is not finitely axiomatisable

2008 ◽  
Vol 73 (3) ◽  
pp. 940-952
Author(s):  
Ian Hodkinson ◽  
Altaf Hussain
Keyword(s):  
Modal Logic ◽  
Affine Plane ◽  
Kripke Frame ◽  
Affine Planes ◽  
Modal Language ◽  
Line Point ◽  
Point Line

AbstractWe consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

The Steiner system S(5,8, 24) constructed from dual affine planes

1986 ◽  
Vol 103 (1-2) ◽  
pp. 147-160
Author(s):  
G. A. Kadir ◽  
J. D. Key
Keyword(s):  
Affine Plane ◽  
Steiner System ◽  
Maximal Subgroups ◽  
Affine Planes ◽  
Tactical Configuration ◽  
Dual Affine ◽  
Mathieu Groups

SynopsisWe construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.

scholarly journals A New Lower Bound for the Size of an Affine Blocking Set

10.37236/7827 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Maarten De Boeck ◽  
Geertrui Van de Voorde
Keyword(s):  
Lower Bound ◽  
Affine Plane ◽  
Blocking Set ◽  
Blocking Sets ◽  
Affine Planes ◽  
Set Of Points

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geqslant 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

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