Central Collineations Research Articles

Central Collineations in Fuzzy and Intuitionistic Fuzzy Projective Planes

In this paper, the fuzzy counterparts and intuitionistic fuzzy counterparts of the central collineations defined in classical projective planes are defined in fuzzy and intuitionistic fuzzy projective planes, respectively. The properties of fuzzy and intuitionistic fuzzy projective planes left invariant under the fuzzy central collineations and the intuitionistic fuzzy central collineations are characterized depending on the base point, base line and the membership degrees of fuzzy projective plane and intuitionistic fuzzy projective plane.

Domesticity in Generalized Quadrangles

An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4.

A family of flat Laguerre planes of Kleinewillinghöfer type IV.A

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillinghofer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillinghofer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillinghofer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.

Planes and processes

This article presented to Combinatorics 2006 is a survey of finite projective planes and the processes used to construct them. All non-translation planes are described, fundamental processes in translation planes are defined and some of these are used to connect semi-field flocks with symplectic spreads. Hermitian ovoids are connected to extensions of derivable nets, and three types of ‘lifting’ methods are discussed. Furthermore, hyperbolic fibrations and ‘regulus-inducing’ central collineation groups are connected to flocks of quadratic cones. Finally, hyper-reguli and multiple hyper-regulus replacement are considered.

Central Automorphisms of Veblenian Nearaffine Planes

The paper deals with nearaffine planes described by H. A. Wilbrink. We consider their central automorphisms, i.e. automorphisms satisfying the Veblen condition, which become central collineations in connected projective planes. Moreover, a concept of c

THEOREMS ON PRODUCTS OF CENTRAL COLLINEATIONS WITH DISTINCT CENTERS OR AXES APPLIED TO THE BENZ PLANES

THEOREMS ON PRODUCTS OF CENTRAL COLLINEATIONS WITH DISTINCT CENTERS OR AXES APPLIED TO THE BENZ PLANES

Ovoids of PG(3, q) stabilized by a central collineation

In this paper we give a short proof that an ovoid of PG(3, q) is stabilized by a non-trivial central collineation of PG(3, q) if and only if it is an elliptic quadric.

M‐Parallelisms

Generalizations of the Johnson parallelisms are given using an index m subgroup of a Pappian central collineation group. The parallelisms, called m‐parallelisms, are constructed and the isomorphisms classes are discussed.

The Decomposition of Any Projectivity of the Projective Plane Onto some Special Central Collineations

The Decomposition of Any Projectivity of the Projective Plane Onto some Special Central Collineations

Some remarks on an article of F.S.Cater

The article [1] contains, among others, certain theorems concerning the problem of decomposing of any projective transformation onto central collineations with given axes or poles. In tis paper some of these theorems are corrected and synthetic proofs of them are given.

Connected 4-dimensional stable planes with many central collineations

This paper gives a complete classification of all connected 4-dimensional stable planes with the property that every point is the centre of a non-trivial central collineation. It is shown that under these assumptions the automorphism group has an open orbit on the point space. This implies the existence of an open subplane that carries the additional structure of a generalized symmetric space in the sense of differential geometry. Now the classification of all 4-dimensional generalized symmetric planes yields the desired classification.

Projective collineations as products of homologies, elations, and projective reflections

A Desarguesian geometry can be described by a vector space over a (not necessarily commutat ive) field. The geometry is Pappian if and only if the field is commutative. Projective collineations can be described by linear transformations. Every projective collineation of a Desarguesian projective geometry of dimension n is a product of at most n + 1 central collineations. This is a consequence of a result of R. Baer (cf. [1], Proposition 1, p. 65). K. W. Gruenberg and A. J. Weir prove a similar theorem for Pappian geometries ([7], Theorem 8, p. 58)? ) O. Veblen and J. W. Young [12], G. Hessenberg [9], and G. Pickert ([10], p. 113) deal with the case n = 2. We shall deal with Desarguesian geometr ies of any, even infinite, dimension. We shall see that a projective coUineation p can only be factored into central collineations if the codimension of the fix of p, codim F(p) , is finite (cf. Lemma 4). We shall turn our attention to a refinement of the result in [1], that means, we shall determine for each projective collineation the minimal number of factors required (cf. Theorem 6). This task is known as the length problem. The length problem has different answers in Pappian and Desarguesian

Central collineations and the parallel axiom in stable planes

Central collineations and the parallel axiom in stable planes

Kollineationen und Schliessungssätze für Ebene Faserungen

Every affine central collineation of a translation plane π induces a special collineation of the projective space π spanned by the spreadF belonging to π. Here the relations between these special collineations of π and certain incidence propositions inF are investigated; so new proofs are given for some characterisations of (A,B)-regular spreads included in [7].

On central collineations of derived semifield planes

LetG denote the collineation group generated by the set of all affine central collineations in a derived semifield plane. We present a characterization of the Hall planes in terms of the order ofG. This essentially allows the extension of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semifield planes. That is, a derived semifield plane of order q2 is a Hall plane precisely when it admits q+1 involutory central collineations.

Products of central collineations

An n by n matrix M over a (commutative) field F is said to be central if M − I has rank 1. We say that M is an involution if M 2= I; if M is also central we call M a simple involution. We will prove that any n-by- n matrix M satisfying det M=±1 is the product of n+2 or fewer simple involutions. This can be reduced to n+1 if F contains no roots of the equation x n =(−1) n other than ±1. Any ordered field is of this kind. Our main result is that if M is any n-by- n nonsingular nonscalar matrix and if x i ∈ F such that x 1⋯ x n =det M, then there exist central matrices M i such that M= M 1⋯ M n and x i =det M i for i=1,…, n. We will apply this result to the projective group PGL( n,F) and to the little projective group PSL( n,F).

On translation planes with affine central collineations

On translation planes with affine central collineations

A homomorphism theorem for projective planes

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

A class of translation planes admitting elations which are not translations

We are interested in the extent to which finite t ranslat ion planes can be characterized by the central collineations, other than those with axis 1r162 which they admit . While the class of known translat ion planes is ve ry large, there still is a shortage of kinds of examples. Lff~C~BU~G [2] has shown t h a t in the case of the planes associated with the Suzuki groups, each affine line is the axis of a non-trivial elation. There have been no known examples of t ranslat ion planes of odd order in which more than one point at infinity is on the axis of a non-trivial elation which is no t a translation. The construct ion we give here gives examples of planes with this property. We have shown in [5] tha t , for characteristic greater t han 3, two elations of a t ranslat ion plane which are not translations and have different centers mus t generate a group isomorphic to S L (2, u) for some u. I n our case, the eollineation group has a subgroup isomorphic to GL (2, u) and contains homologies as well as translations. Essentially the same construct ion gives examples of planes which are no t semifield planes in which a single point at infinity is the center of elations which are not translations.

The Collineation Group of the Veblen-Wedderburn Plane of Order Nine

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.