Finite collineation groups in projective spaces of one two and three dimensions

Abstract

The account given in this thesis of the finite collineation groups in projective spaces of one, two and three dimensions is divided into two main sections. Section I indicates some of the methods available for determining the orders of the primitive groups in these spaces, with particular reference to the work done in one and two dimensions by H.F. Blichfeldt, and in three dimensions by G. Bagnera. Section II is an investigation of some of the primitive groups in three dimensions which are generated by biaxial homographies. This latter section has four main paragraphs; in paragraph II the groups generated by biaxial homographies which leave fixed a quadric are determined, and we are concerned with those groups which are isomorphic with symmetric groups in paragraph III; the methods used in these two paragraphs are my own. The group of order 11520 which leaves fixed the Klein 60 15, configuration is the subject of paragraph IV, the operations of this group and some of its subgroups are found by methods based on those used by J. Todd to determine a simple group of order 25920 in four dimensions. Similar methods are used in paragraph V to find the operations of a simple group of order 25920 leaving fixed a configuration of forty-five points and planes; the configuration has been described by G. Bagnera, but I can find no other account of the group in three dimensions.