Cross Ratios of Points and Lines in Moufang Planes
نویسندگان
چکیده
In this paper, first we extend the known definition of cross-ratio of collinear points to whole Moufang plane. Later we introduce the cross-ratios for lines and the known results about the cross-ratios of points which are adapted to crossratios of lines without using the principle of duality. Finally, we give a theorem which describes the relation between the cross-ratios of points and lines. Mathematics Subject Classification: 51A35
منابع مشابه
Moufang Quadrangles of Mixed type
In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual net associated to the regular points satisfy the Axiom of Veblen-Young, or a very we...
متن کاملQuadrangles embedded in metasymplectic spaces
During the final steps in the classification of the Moufang quadrangles by Jacques Tits and Richard Weiss a new class of Moufang quadrangles unexpectedly turned up. Subsequently Bernhard Mühlherr and Hendrik Van Maldeghem showed that this class arises as the fixed points and hyperlines of certain involutions of a metasymplectic space (or equivalently a building of type F4). In the same paper th...
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 کاملSolution of a Question of Knarr
The Moufang condition is one of the central group theoretical conditions in Incidence Geometry, and was introduced by Jacques Tits in his famous lecture notes (1974). About ten years ago, Norbert Knarr studied generalized quadrangles (buildings of Type B2) which satisfy one of the Moufang conditions locally at one point. He then posed the fundamental question whether the group generated by the ...
متن کاملMoufang sets related to polarities in exceptional Moufang quadrangles of type F 4
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...
متن کامل