Compact disconnected planes,inverse limits and homomorphisms

Every compact disconnected projective plane can be written as an inverse limit of finite discrete incidence structures. Every finite projective plane is a continuous epimorphic image of some compact disconnected projective plane. There exist compact disconnected projective planes of Lenz type V which do not admit any continuous epimorphism onto a finite projective plane.Supported in part by a travel grant of the Deutsche Forschungsgemeinschaft.     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

基于有限射影平面上(ω,r,d)-CFF(N,T)系统的构作

在有限射影平面上利用有限射影平面的性质构作了(ω,r,d)-CFF(N,T)系统,并利用有限射影平面的性质计算了它的参数.最后利用一个有限点集构作了一个(ω,r,d)-DS(N,T)系统并计算了它的参数.     

A new proof of a theorem of Dembowski

It was shown by P.Dembowski [1;Satz 3] that any finite semiaffine plane(=FSAP) is of the types:

1. A finite affine plane,
2. A finite projective plane with one line and all its points except one deleted,
3. A finite projective plane with one point deleted,
4. A finite projective plane.

This was established by using the results obtained for natural parallelisms of incidence structures. The purpose of this note is to give a new proof based on purely combinatorial arguments.     

Classification of linear spaces without three mutually parallel lines

We classify all finite linear spaces without three mutually parallel lines. Apart from two exceptions, such a space is necessarily a generalized projective plane, a simple extension of a generalized projective plane, or a complete inflated affine plane with a generalized projective plane at infinity.Dedicated to Giuseppe Tallini on the occasion of his 60^th birthday     

On automatic presentations of projective planes

We study automatic presentations of projective planes and prove that no freely generated projective plane has any automatic presentations. Every desarguesian (pappian) projective plane is shown to be automatically presentable if and only if it is finite.     

Conics in finite projective planes

There does not exist a general theory of conics in finite projective planes, because the many definitions of conics which are equivalent in desarguesian projective planes yield different types of conics in more general situations. Thus even the use of the word conic can lead to confusion, particularly in the finite case. This note is an attempt to clarify these various definitions and give as an example in a finite projective plane a von Staudt conic which is not an Ostrom conic. We conjecture that any finite projective plane admitting an Ostrom conic must be desarguesian.     

On tangentially transitive projective planes

The existence of Baer collineations in a projective plane is related to the existence of desargues-like configurations. The plane of order four is characterized as the only finite plane that possesses a Baer subplane partition into tangentially transitive Baer subplanes which is preserved by each of the tangentially transitive groups. It is shown that a finite projective plane has either no or one tangentially transitive Baer subplane or is partially transitive of Hughes type (4, m), (5, m) or (6, m) for some m. The Lenz-Barlotti classes which contain a finite plane which is not a translation plane nor its dual and which possesses a tangentially transitive Baer subplane are shown to be classes I.1 and II.1.     

Characterization of projective planes of small prime orders

The method developed recently by the author to handle collineation groups of relative small order is used here to characterize the Desarguesian projective plane of order 11 by a collineation group isomorphic to the symmetric group of 4 letters. Also we prove that a finite projective plane of order 11, 13, or 17 is Desarguesian if it admits a collineation group, which is either strongly irreducible or non-abelian simple.     

Rings Associated with Finite Projective Planes and Their Isomorphisms

We announce an explicit form of the standard basis for the 2-extended ring associated with the cellular ring generated by the incidence graph of a finite projective plane. This enables us to give the first example of a distance-regular graph that satisfies the 6-condition and is not distance-transitive. One more consequence of the result obtained is that the cellular rings of any two projective planes of the same order are 2-isomorphic. This implies that, if for some order there exist at least two nonisomorphic and nondual to each other projective planes, then the separability number of the cellular ring of any projective plane of this order is greater than or equal to 3 and, moreover, it is equal to 3 for a Galois plane. Bibliography: 9 titles.     

Planar,Toroidal, and Projective Commuting and Noncommuting Graphs

In this paper, all finite groups whose commuting (noncommuting) graphs can be embed on the plane, torus, or projective plane are classified.     

若干单群与射影平面直射群

设G是有限群,H(?)G。如果H≌~2B_2(q)或H≌~2G_2(q)或H≌PSU(3,q),则G不与任何射影平面的点传递直射群同均。本文对以下问题给出了一般方法:证明以某些几乎单群为点传递自同构群的线性空间不是射影平面。     

Configurations arising from maximal arcs

We show that the existence of a maximal arc in a finite projective plane implies the existence of certain block designs and partial geometries. The block designs obtained have interesting resolvability properties.     

Hyperovals in the known projective planes of order 16

We construct by computer all of the hyperovals in the 22 known projective planes of order 16. Our most interesting result is that four of the planes contain no hyperovals, thus providing counterexamples to the old conjecture that every finite projective plane contains an oval. © 1996 John Wiley & Sons, Inc.     

Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].     

Sets of type (0,n) in Steiner systems and in finite projective planes

In this paper we study the sets of type (0,n)₁ in a finite projective plane and the inclusion relations between them.     

Characterization of orthomaps on the Cayley plane

We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns’s theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions.     

Closure planes

We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a “root” plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz–Barlotti types for these planes are then linked to the Lenz–Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz–Barlotti types II.1 and V.1.     

Isotopy and geotopy for ternary rings of projective planes

We investigate the question of when two ternary rings coordinatize the same finite projective plane. Necessary and sufficient conditions are obtained for right quasifields and for right distributive linear rings whose multiplicative loop is a group.