A theorem in finite projective geometry and some applications to number theory

TL;DR: In this article, it was shown that there is always at least one collineation of period q with respect to any point in the projective plane PG(2, pn) for every prime p and positive integer n.

Abstract: A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let