Three Ways of Measuring Distance, Three Orbits, Three Subdegrees, or the Great Theorems of Cameron and Kantor, Volume 49, Number 7

Finite Universe Is our universe finite or infinite? As we travel further and further will our journey ever end or must we go on forever? The second possibility being frightening, it is convenient to assume the first. In other words, let us postulate that we live in the projective three-space. The quantum theorists have told us that matter, instead of being continuous, is discrete, so our universe has a finite number of points. The physicists also like to include time as the fourth dimension, and to represent further states, they even like the dimension to be any positive integer. So, as a thought experiment, we might as well suppose our universe U to be a finite projective space of any positive dimension N. Thus the universe U consists of a finite number of points. Some points are collinear, i.e., they lie on a line. Any two points determine a line. Any three noncollinear points determine a plane. Any four noncoplanar points determine a solid. And so on. Finally N + 1 points, not lying on any (N − 1) dimensional subspace, span the whole space U. An (N − 1)-dimensional subspace of U is called a hyperplane. Those properties of U which do not change when we change coordinates are more interesting than those which change. Making a coordinate change amounts to applying a collineation to U. A collineation is a bijection, i.e., a one-to-one onto map, of U onto itself which preserves collinearity, i.e., it sends collinear points to collinear points. We can follow up a collineation g by a collineation h to give us their composition hg . Also reversing g gives us its inverse g−1. The set of all collineations of U thus forms a group Γ (U ). A perspectivity is a collineation which fixes all points on a hyperplane. It will then also fix all hyperplanes through a certain point. If that point is on that hyperplane, then the perspectivity is an elation, otherwise a homology; see the 1920 projective geometry book by Veblen and Young. The subgroup of Γ (U ) generated by all elations is the little projective group denoted by S(U ) . Likewise the group generated by all perspectivities is the full projective group G(U ). It is easily checked that the subgroup S(U ) is doubly transitive on U, i.e., any two points of U can be sent to any other two points of U by a suitable member of S(U ) . One of the great theorems of Cameron-Kantor, Theorem I of their 1979 paper [6], says that conversely if a subgroup of the collineation group Γ (U ) is doubly transitive on U, then it must contain S(U ) , with a few small exceptions which we shall mention in a moment.