Periodic Linear Transformations of Affine and Projective Geometries

Extract

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pⁿ), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pⁿ), admits a collineation L of period q = (p^sn — 1)/ (pⁿ — 1). Since this q is the number of points of PG(s — 1, pⁿ), Singer's result states that the points of PG(s — 1, pⁿ) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pⁿ) which defines a PG(s — 1, pⁿ).”