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.     

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.Dedicated to Professor H. Salzmann on his 60th birthday     

Some collineations in projective planes induced by net collineations

When in a 3-net the three line pencils are permuted, then with some additional requirements the net is mapped onto itself in a way which also induces a map of a coordinatizing loop (Q^*), onto another, isostrophic loop (Q^*, Ø). Every identity ab = aØb induces a net collineation and, simultaneously, a loop law. This procedure is used for finding collineations of a translation plane such that their existence is a necessary and sufficient condition for the validity of some laws in the multiplicative loop of a coordinatizing quasifield. Proofs can be found in [1, 2, 3].     

Loop laws and collineations of planes

In translation planes, collineations are determined whose existence is necessary and sufficient for the validity of certain laws in the multiplicative loop of the coordinatizing quasifield. A general method is developed for loop laws derived from loop isostrophisms. Dedicated to the memory of Professor Eri Jabotinsky     

Polarities and planar collineations of Moufang planes

We show that conjugacy classes of Baer involutions and non-elliptic polarities, respectively, of proper (i.e., non-desarguesian) Moufang planes are interrelated. Restriction of the conjugating group to the stabilizer of a triangle or a quadrangle does not refine the classes. These results are applied to prove transitivity properties for the centralizers of these polarities. Along the way, a new proof is obtained for the fact that the automorphism group of a Moufang plane acts transitively on quadrangles.     

Non-existence of collineations equivalent to some loop laws

If in a translation plane there is a set of collineations whose existence is necessary and sufficient for the validity of the multiplicative left-inversive law, the weak-inversive law, the anti-automorphicinverse law, or the left Bol law in one of the coordinatizing right quasifields, then the plane is a Moufang plane. The same holds also for the Moufang law, if the coordinatizing quasifield is not the Hall system of order 9.     

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.     

On collineations and dualities of finite generalized polygons

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation θ of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by θ. As a special case we consider generalized 2n-gons of order (1, t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t.