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SEMILINEARTRANSFORMATIONS OVER FINITE FIELDS ARE FROBENIUS MAPS

Published online by Cambridge University Press:  01 May 2000

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

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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 42 , Issue 2 , May 2000 , pp. 289 - 295
Copyright
2000 Glasgow Mathematical Journal Trust