A Unified Theory of Projective Spaces and Finite Abelian Groups

The similarity between finite-dimensional projective spaces and finite abelian groups has often been noted(1); and thus one may expect that the more general features of these two theories are identical. But the likeness is more than a superficial one; and consequently it is possible to give a unified treatment for spaces and groups. It may be worthwhile to indicate in a few lines the developments leading up to a scientific situation that made such a joint theory as we are offering here a possibility. The evolution of geometrical thought pertinent to our problem is perhaps best described by two textbooks: Bocher's Higher Algebra which exposed the identity of geometry and the theory of linear equations; and Veblen and Young's Projective Geometry whose presentation of the theory broke down the restriction to the two geometries over the real and the complex number field; and enlarged the domain to be considered to the projective geometries over any sort of field, whether finite or infinite, commutative or not. Any further progress had to be a progress in the theory of linear equations; and this was found in the treatment of the theory without using determinants—a concept that had to be thoroughly debunked to make these (and other) extensions possible. This was the starting point for further generalizations, generalizations in a direction that is different from ours—notably the theory of linear equations with infinitely many variables and in particular its geometrical counterpart, J. von Neumann's continuous geometry. In the theory of finite abelian groups only one generalization was needed. We refer to the extension of this theory by introducing the concept of an abelian group which admits operators from some given ring or some other domain. For this concept makes it possible to consider projective geometry a—rather special—chapter in the theory of abelian groups, since the n-dimensional projective space over the (not necessarily commutative) field F of coordinates is nothing but the set of F-admissible subgroups of an abelian group of rank »+1 over F and the linear forms over this geometry are just the characters of this underlying group. Our problem is now easily stated: to characterize a class of abelian operator groups which comprises both the