Theorems on Products of Central Collineations With Distinct Centers or Axes Applied to the Benz Planes

Abstract

Abstract. There are three kinds of the Benz planes: Möbius planes, Laguerre planes and Minkowski planes [2,3,7]. In any Benz plane an automorphism y is central if ¡p has a fixed point P and becomes a central collineation in the projective derived plane induced by P. Such central automorphisms have been considered by many authors (cf. [8,13,11,12,10]), in particular the automorphism groups were classified. Usually product of two central collineations without common center or common axis is not central. But in some special cases it is central [4]. In this paper we apply theorems concerning such special cases -to the Benz planes. IntroductionFoundations of the Benz planes are given in [2]. The first chapter of [2] is devoted to three kinds of classical models inspiring three kinds of the Benz planes. The models realized on three projectively non equivalent quadrics (sphere, cylinder and one-sheeted hyperboloid) seem to be most popular (for details of those and other models see, e.g. [2, pp. 1-4; 9-21; 42-81] and [7, footnote, p. 45]). Here the point set coincides with the point set of the quadric and all its non degenerate plane sections are taken as blocks. Roughly speaking a Benz plane is some structure B = (II, A) with the point set II provided with some family A of subsets of II called circles. Except circles further families (one or two) of subsets of II, called generators, may be given. Each family of generators is a partition of II and two distinct points are on a common generator iff they are not on a common circle. The Benz plane is called a Möbius plane, a Laguerre plane or a Minkowski plane according as it has 0, 1 or 2 families of generators. For every point P of the Benz plane ¡3 = (II, A) the internal structure Bp [4, p. 3] (resp. Bp provided 1991 Mathematics Subject Classification: 51A35, 51B10, 51B15, 51B20, 51E15, 12J15.

On some operations on finite affine planes applied to the theory of regular configurations

“…As it was proved in [5] and [18] (see also [15]), the Benz-plane induces in each of its points a derived affine plane by means of an operation the result of which will be called a chain contraction (also in the case when the (p + 2)-tuple β p (for p > 2) replaces the Benz-plane). By creating a chain contraction of the system β p at a point A, we remove all generators containing A and all chains disjoint with A.…”

“…The systems β 0 and β p (for p = 1, 2) satisfying some conditions are the Möbius, Laguerre and Minkowski planes, 168 H. Makowiecka respectively. We shall use the axiomatic description of the Benz plane ( [15], p. 639-643), which takes into account earlier books and papers ( [3], [5], [18]). …”

“…In the case of the Benz plane we obtain an affine plane, the so-called derived affine plane at a point A (cf. [15], [26] or [5]). …”