Projectivities In Projective Planes

Abstract

In section A the question, under what conditions a projectivity of a line — defined as a product of perspectivities — with 3 fixed points is the identity (condition P3), leads to the condition of Desargues and Pappos, the Hessenberg Theorem (“Pappos” implies “Desargues”) and the Hessenberg Theorem (“Pappos” implies “Desargues”) and the Fundamental Theorem of Projective Geometry, giving several conditions equivalent to P3; historical notes to these developments are provided in B. Section C proves the Schleiermacher Theorem: If every projectivity of a line with 5 fixed points is the identity, then the Pappos condition holds. Also consequences of the similar condition with 6 fixed points are considered. In D the Moufang planes are characterized by the existence of a permutation group on a line, sharply transitive on 1\{P} for a point P on 1, and normalized by those projectivities of 1 onto itself with fixed point P (Generalized Luneburg-Yaqub- Theorem).