Collineations in a Finite Projective Geometry

m *'* aiixr + ai2< + ■ ■ • + ai*+i°€li fi_, h, xk+i ak+nxi ~r an-i2a:2 + ■+ a*+i*+ia;i:+i where wi is zero or an integer less than n, is a collineation. As a result of Dr. Levi's article J it is possible to prove the converse proposition, namely, that every collineation in PG(k,pn) is of type (1). The following argument connects this theorem directly with our former article. It will be sufficient to give the argument for the case, k = 2. A projective collineation, § or, in other words, a linear transformation, is determined by the quadrangle into which it transforms the quadrangle (0 0 1), (111), (Oil), (101). It follows that an arbitrary collineation is the product of a projective collineation by a collineation which leaves these four points invariant. The latter type of collineation may be called antiprojective, in the language of Segre. || Its existence is proved by the existence of transformations of type (1) where m +0. An antiprojectivity, by its definition, leaves invariant the points of the xx axis for which xx/x3 is 0, 1, or oo, and, therefore, all points of the chain determined by these three, namely, all points whose coordinates are integers modulo p. From the quadrangle-construction for addition and multiplication and the fact that a collineation transforms a complete quadrangle into a complete quadrangle it follows that if three points of the xx axis for which xx/xi is a, 6, and c respectively are so related that