Elations and homologies in collineation groups of finite projective planes

Let G be a collineation group of a finite projective plane. The action of G on the centers and axes of non-identity elations and homologies is discussed. There are several results on the possible numbers of orbits of centers, axes, and center-axis pairs of homologies and elations of a particular order. Several results on the generation of homologies or elations by other homologies or elations reveal additional information on the structures formed by the centers and axes. Some sets of sufficient conditions for the centers and axes to form Desarguesian subplanes are given.     

Subplanes of finite projective planes

We study questions of the hereditariness of the property of (, )-transitivity (the property of being a plane of translations) of a finite protective plane for its subplanes. In particular it is shown that hereditariness holds for subplanes containing the point .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1101–1102, July–August, 1991.     

Elations in four-dimensional planes

Let K and L be two distinct lines of a topological projective plane which is homeomorphic to the complex plane. If for each center cϵK there exists a homology ≠1 with axis L, then the elation group with center K ∩ L and axis L is transitive.     

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.     

Configurations in finite projective planes

We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1345–1359, October, 1997.     

On homologies of finite projective planes

Study of finite projective planes which have a polarity preserved by many homologies. This research was supported in part by NSF Grant GP-33223.     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

Some results on finite projective planes

This work contains a survey of some basic results on finite projective planes which were obtained by a group of mathematicians from Krasnoyarsk since 1985. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 63, Algebra-13, 1999.     

On flag collineations of finite projective planes

s paper studies collineation groups of a finite projective plane containing flag collineations. Among other results, a characterization of a finite Desarguesian projective plane is given.Partially supported by grants from CNPq do Brasil and NSERC of Canada.     

Upper chromatic number of finite projective planes

For a finite projective plane , let denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is , which is tight apart from a multiplicative constant in the third term :

• (1) As holds for every projective plane of order q.
• (2) If q is a square, then the Galois plane of order q satisfies .

Our results asymptotically solve a ten‐year‐old open problem in the coloring theory of mixed hypergraphs, where is termed the upper chromatic number of . Further improvements on the upper bound (1) are presented for Galois planes and their subclasses. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 221–230, 2008     

Pasch geometric spaces inducing finite projective planes

Summary A geometric space over a geometric sfield of dimension three induces a protective plane. A relation between the order of the projective plane and that of the geometric sfield is obtained. For a particular order of the sfield, the induced projective plane is shown to be desarguesian.