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Rational rigidity for ${E}_{8}(p)$

Published online by Cambridge University Press:  17 July 2014

Robert Guralnick
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA email guralnic@usc.edu
Gunter Malle
Affiliation:
FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany email malle@mathematik.uni-kl.de
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Abstract

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We prove the existence of certain rationally rigid triples in ${E}_{8}(p)$ for good primes $p$ (i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

Type
Research Article
Copyright
© The Author(s) 2014 

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