We determine all homogeneous pseudo-embeddings of the affine space AG(n, 4) and the projective space PG(n, 4). We give a classification of all pseudo-hyperplanes of AG(n, 4). We also prove that the two homogeneous pseudo-embeddings of the generalized quadrangle Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable.

In this paper, we classify the finite generalized quadrangles of order (s, t), s, t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L , M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points.

Introduction: The Fortuna Tessera quadrangle (V–2, Fig. 1) is in the northern hemisphere of Venus and extends from 50° to 75° N. and from 0° to 60° E. In a regional context, this territory represents a large region of tessera terrain [1] that includes the major portion of Fortuna Tessera in the northern half of the map area and Laima Tessera in the SE corner of the quadrangle. The tessera regio...

To Professor Helmut Wielandt, to commemorate his seventy-fifth birthday 1. Introduction In [2] a new generalized quadrangle with parameters q2, q was constructed using the G 2(q) generalized hexagon (q == 2 (mod 3), q > 2). In addition, an elementary group theoretic technique was presented for constructing generalized quadrangles. This technique was refined by Payne [3] in order to simplify cal...

Desarguesian projective planes of square order are characterized by the property that every quadrangle lies on a unique Baer subplane.

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Mouf...

Let & = (P, £, I) be a generalized quadrangle of order 5, s φ 1, having a flag (x, L) with χ and L regular. Then a generalized quadrangle SP = (Ρ',Β',Ι') of order s can be constructed. We say that &" is obtained by switching from & with respect to (je, L). Examples are given where ¥ £ &'\ e.g., starting from a T2(O) of Tits, with O an oval of PG(2, q), q even, with nucleus «, the GQ T2((O {*}) ...

Suppose S is a finite generalized quadrangle (GQ) of order (s, t), s 6= 1 6= t, and suppose that L is a line of S. A symmetry about L is an automorphism of the GQ which fixes every line of S meeting L (including L). A line is called an axis of symmetry if there is a full group of symmetries of size s about this line, and a point of a generalized quadrangle is a translation point if every line t...

Using concepts from valuation theory, we obtain a characterization of all collinearity-preserving functions from one affine or projective Desarguesian plane into another. The case in which the planes are projective and the range contains a quadrangle has been treated previously in the literature. Our results permit one or both planes to be affine and include cases in which the range contains a ...