Remarks on projectivities

Aequationes mathematicae -     

Linear representation of a class of projective planes in a four dimensional projective space

Summary The theory of linear representations of projective planes developed by Bruck and one of the authors (Bose) in two earlier papers [J. Algebra1 (1964), pp. 85–102 and4 (1966), pp. 117–172] can be further extended by generalizing the concept of incidence adopted there. A linear representation is obtained for a class of non-Desarguesian projective planes illustrating this concept of generalized incidence. It is shown that in the finite case, the planes represented by the new construction are derived planes in the sense defined by Ostrom [Trans. Amer. Math Soc.111 (1964), pp. 1–18] and Albert [Boletin Soc. Mat. Mex,11 (1966), pp, 1–13] of the dual of translation planes which can be represented in a 4-space by the Bose-Bruck construction. An analogous interpretation is possible for the infinite case. This research was sponsored by the National Science Foundation under Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-68-1406. This research was conducted while the author was visiting professor at the University of North Carolina at Chapel Hill. His research was also partially supported by C.N.R. Entrata in Redazione il 28 maggio 1970.     

The normal fitting class generated by a finite soluble group

Sunto Per ogni gruppo finito risolubile G, la classe di Fitting normale generata da G può essere descritta mediante certe coppie di Fitting introdotte da Laue, Lausch e Pain in [4] e riprese da Berger in [2]. Si ottiene in particolare una caratterizzazione piuttosto semplice della classe di Fitting normale generata da S₃.     

Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

On the definition of baer subplanes of infinite planes

LetP be an infinite projective plane and letB be a subplane ofP. An example is given which shows that the following two conditions are independent: (1) Every line ofP is incident with at least one point ofB. (2) Every point ofP is incident with at least one line ofB.This research was supported by the C.N.R. (Italy).     

Multigroups and the foundations of geometry

The aim of this paper is to show that the foundations of geometry and the theory of moultigroups are strictly related. Dedicated to Helmut Salzmann on his 60^th birthady.