Glasgow Mathematical Journal



SEMILINEAR TRANSFORMATIONS OVER FINITE FIELDS ARE FROBENIUS MAPS


U. DEMPWOLFF a1, J. CHRIS FISHER a2 and ALLEN HERMAN a2
a1 FB Mathematik, Universita¨t Kaiserslautern, Postfach 3049, D-67653 Kaiserslautern, Germany. E-mail: dempwolff@mathematik.uni-kl.de
a2 Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada, S4S 0A2. E-mail: fisher@math.uregina.ca, aherman@math.uregina.ca

Abstract

In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

(Received October 27 1998)