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.