James Singer  has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, p
n). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, p
n). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.