Collineation groups containing homologies

Abstract

In this paper we examine collineation groups which contain homologies of finite projective spaces. The first two theorems give minimal conditions for a collineation group containing homologies of a finite desarguesian projective space to contain its little projective group. In Theorem 1 we prove that if fl is a collineation group generated by some homologies of a finite desarguesian projective space Hk such that (a) no subspace of Hk is left fixed by Ii’ and (b) some hyperplane is the axis of homologies in 17 for more than one center, then D contains the little projective group of Hk . Then, as a consequence of Theorem 2, we show that for finite desarguesian projective planes conditions (a) and (b) are sufficient for 17 to contain the little projective group. However, for It > 3, it is definitely necessary to assume that Ii’ is generated by the homologies. In Theorem 3 we show that if H, is a finite projective plane (not necessarily desarguesian) and if Ii’ is a collineation group of H2 fixing no point or line such that (a) some line 8 is the axis of homologies in 17 for more than one center and (b) some point P EL’is the center of a homology in n, then all the centers and axes of the homologies in 17 form a desarguesian subplane, Hi say, of H, and 17 restricted to Hi contains its little projective group. ‘rhis result agrees with the result for desarguesian planes and, hence, is the best possible. Finally in Theorem 4 we examine the possibilities for 17 when every point of Hk is the center of a homology in II. We prove, almost as a corollary to Theorems 1 and 3, that if no subspace is left fixed by Ll, then Hk is desarguesian and 17 contains its little projective group.