Split Semi-Biplanes in Antiregular Generalized Quadrangles
نویسندگان
چکیده
There are a number of important substructures associated with sets of points of antiregular quadrangles. Inspired by a construction of P. Wild, we associate with any four distinct collinear points p, q, r and s of an antiregular quadrangle an incidence structure which is the union of the two biaffine planes associated with {p, r} and {q, s}. We investigate when this incidence structure is a semi-biplane.
منابع مشابه
A geometric proof of a theorem on antiregularity of generalized quadrangles
A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular.
متن کاملAutomorphisms and characterizations of finite generalized quadrangles
Our paper surveys some new developments in the theory of automorphisms and characterizations of finite generalized quadrangles. It is the purpose to mention important new results which did not appear in the following standard works (or surveys) on the subject: Collineations of finite generalized quadrangles (S. E. Payne, 1983), Finite Generalized Quadrangles (S. E. Payne and J. A. Thas, 1984), ...
متن کاملA class of functions and their application in constructing semi-biplanes and association schemes
We give an alternative proof of the fact that a planar function cannot exist on groups of even order. The argument involved leads us to define a class of functions which we call semi-planar. Through the introduction of an incidence structure we construct semi-biplanes using semi-planar functions. The method involved represents a new approach to constructing semi-biplanes and provides infinite c...
متن کاملGeometric Characterizations of Finite Chevalley Groups of Type B2
Finite Moufang generalized quadrangles were classified in 1974 as a corollary to the classification of finite groups with a split BN-pair of rank 2, by P. Fong and G. M. Seitz (1973), (1974). Later on, work of S. E. Payne and J. A. Thas culminated in an almost complete, elementary proof of that classification; see Finite Generalized Quadrangles, 1984. Using slightly more group theory, first W. ...
متن کاملLocal Sharply Transitive Actions on Finite Generalized Quadrangles
We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentago...
متن کامل