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Havliček-Tietze configurations in various projective planes

Jakóbowski J., Kacperek D.

Demonstratio Mathematica

2014

Vol. 47, nr 4

979--988

A. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havliček–Tietze configuration (shortly H - T) in the desarguesian projective plane is equivalent to existence in the associated field, a root of polynomial x2 + x + 1, different from 1. We show that such a configuration exists in every projective plane over Galois field GF(p2) for p ≠ 3. As it has been demonstrated, in a projective plane over arbitrary field F, each hexagon contained in H - T, satisfies the Pappus–Pascal axiom, even if F is noncommutative. Moreover, such a hexagon either is pascalian or has exactly one pair of opposite sides intersecting at a point collinear with two points not belonging to these sides. In particular, all such hexagons are pascalian iff char F = 2. For the (noncommutative) field of quaternions, we have determined the set of all roots of the mentioned polynomial. Every H - T is the special Pappus configuration, in which three main diagonals of the hexagon are concurrent.

Central automorphisms of Veblenian nearaffine planes

Cudna-Lasecka K., Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 3

337-347

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 central pseudo-automorphism is considered, i.e. some bijections in a nearaffine plane are not automorphisms but they become central collineations in the related projective planes.

Theorems on products of central collineations with distinct centers or axes applied to the Benz planes

Jakóbowski J.

Demonstratio Mathematica

2004

Vol. 37, nr 3

639--653

There are three kinds of the Benz planes: Mobius planes, Laguerre planes and Minkowski planes [2, 3, 7]. In any Benz plane an automorphism φ is central if φ 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.