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Relative subgroups in Chevalley groups

Published online by Cambridge University Press:  15 March 2010

R. Hazrat ,
V. Petrov  and
N. Vavilov
R. Hazrat
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, U.K., r.hazrat@qub.ac.uk
V. Petrov
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia, victorapetrov@gmail.com
N. Vavilov
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia, nikolai-vavilov@yandex.ru
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Abstract

We finish the proof of the main structure theorems for a Chevalley group G(Φ, R) of rank ≥ 2 over an arbitrary commutative ring R. Namely, we prove that for any admissible pair (A, B) in the sense of Abe, the corresponding relative elementary group E(Φ,R, A, B) and the full congruence subgroup C(Φ, R, A, B) are normal in G(Φ, R) itself, and not just normalised by the elementary group E(Φ, R) and that [E (Φ, R), C(Φ, R, A, B)] = E, (Φ, R, A, B). For the case Φ = F4 these results are new. The proof is new also for other cases, since we explicitly define C (Φ, R, A, B) by congruences in the adjoint representation of G (Φ, R) and give several equivalent characterisations of that group and use these characterisations in our proof.

Keywords

Chevalley groupselementary subgroupcongruence subgroupcommutator formulas
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 3 , June 2010 , pp. 603 - 618
Copyright
Copyright © ISOPP 2010

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