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Rational rigidity for 
${E}_{8}(p)$
Part of:
Field extensions
Structure and classification of infinite or finite groups
Representation theory of groups
Published online by Cambridge University Press: 17 July 2014
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.
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