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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 originalformulation Lang's theorem referred to a semilinear map on an n-dimensional vectorspace 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 isequivalent by change of coordinates to a map induced by a field automorphism. We provide an elementaryproof of the theorem independent of the theory of algebraic groups and, as a by-product of ourinvestigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classicalgroups and to projective geometry. In the latter application we uncover three simple yet surprisingresults.

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