CLASSICAL GROUPS AND
THE CLASSIFICATION
by RONALD SOLOMON
Syllabus:
A. The Classical Groups
The main goal is to
understand the structure of centralizers of semisimple elements in classical
groups.
1. Definitions and
basic properties of the classical groups. Witt's Theorem.
2. Parabolic subgroups:
the Borel-Tits Theorem. Some centralizers of unipotent elements.
3. Reductive subgroups:
Maximal tori. Centralizers of semisimple elements.
B. The Classification
(The Very Generic Case)
The goal is to give
some of the important ideas of the proof that a ''very generic'' simple
group G (to be defined) is a classical group defined over a field of odd
order.
1. The Signalizer Functor
Method: Proving that involution centralizers in G are reductive.
2. The Lyons-Seitz
Generation Theorem and Aschbacher's Component Theorem for ''very generic''
groups.
3. Recognition Theorems
for Classical Groups.
References:
Free online references such as [2]
or parts 1 and 2 of [5] can be downloaded by clicking on them.
[1] M. Aschbacher,
Finite
Group Theory, Cambridge University Press, Cambridge, 1986.
[2] P.J. Cameron, Notes
on Classical Groups, p.j.cameron@qmw.ac.uk.
[3] R. W. Carter, Finite
Groups of Lie Type, Wiley and Sons, London 1972.
[4] J. Dieudonné,
La
Geometrie des Groupes Classiques, Springer, Berlin 1955.
[5] D. Gorenstein,
R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups,
A.M.S. Mathematical Surveys and Monographs, Volume 40, n. 1,
2,
3, 4, 5, 6.
[6] P. Kleidman and
M. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge
University Press, Cambridge, 1990.
[7] R. Steinberg, Lectures
on Chevalley Groups, Yale University Lecture Notes 1968.
[8] D. E. Taylor, The
Geometry of the Classical Groups, Heldermann Verlag, Berlin, 1992.
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