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Computing Galois cohomology and forms of linear algebraic groups

Haller, Sergei


Originalveröffentlichung: (2005) Zugl.: Eindhoven, Technische Universiteit, Proefschrift, 2005
pdf-Format: Dokument 1.pdf (667 KB)

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Freie Schlagwörter (Deutsch): Gruppentheorie , Lineare algebraische Gruppen , Cohomologie , Computeralgebra
Freie Schlagwörter (Englisch): group theory , linear algebraic groups , cohomology , computer algebra
Universität Justus-Liebig-Universität Gießen
Institut: Faculteit Wiskunde en Informatica (Technische Universiteit Eindhoven); Mathematisches Institut der Justus-Liebig-Universität Giessen
Fachgebiet: Externe Einrichtungen
DDC-Sachgruppe: Mathematik
Dokumentart: Buch (Monographie)
ISBN / ISSN: 90-386-0664-8
Sprache: Englisch
Erstellungsjahr: 2005
Publikationsdatum: 11.11.2005
Kurzfassung auf Englisch: We design and implement algorithms for computation with groups of Lie
type. Algorithms for element arithmetic in the Steinberg presentation of
untwisted groups of Lie type, and for conversion between this presentation
and linear representations, were given in [12] (building on work of [15]
and [26]). We extend this work to twisted groups, including groups that
are not quasisplit.


A twisted group of Lie type is the group of rational points of a twisted
form of a reductive linear algebraic group. These forms are classified by
Galois cohomology. In order to compute the Galois cohomology, we develop a
method for computing the cohomology of a finitely presented group $\Gamma$
on a finite group $A$. This method is of interest in its own right. We
then extend this method to the Galois cohomology of reductive linear
algebraic groups.


We give algorithms for computing the relative root system of a twisted
group of Lie type, the root subgroups, and the root elements, as well as
algorithms for the computing of relations between root elements.


As an application, we develop an algorithm for computing all twisted
maximal tori of a finite group of Lie type. The order of such a torus is
computed as a polynomial in $q$, the order of the field $k$. We also
compute the orders of the factors in a decomposition of the torus as a
direct product of cyclic subgroups.