Linear Algebraic Groups and Finite Groups of Lie Type: (Series Number 133 Cambridge Studies in Advanced Mathematics)

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

Table of Contents:
Preface; List of tables; Notation; Part I. Linear Algebraic Groups: 1. Basic concepts; 2. Jordan decomposition; 3. Commutative linear algebraic groups; 4. Connected solvable groups; 5. G-spaces and quotients; 6. Borel subgroups; 7. The Lie algebra of a linear algebraic group; 8. Structure of reductive groups; 9. The classification of semisimple algebraic groups; 10. Exercises for Part I; Part II. Subgroup Structure and Representation Theory of Semisimple Algebraic Groups: 11. BN-pairs and Bruhat decomposition; 12. Structure of parabolic subgroups, I; 13. Subgroups of maximal rank; 14. Centralizers and conjugacy classes; 15. Representations of algebraic groups; 16. Representation theory and maximal subgroups; 17. Structure of parabolic subgroups, II; 18. Maximal subgroups of classical type simple algebraic groups; 19. Maximal subgroups of exceptional type algebraic groups; 20. Exercises for Part II; Part III. Finite Groups of Lie Type: 21. Steinberg endomorphisms; 22. Classification of finite groups of Lie type; 23. Weyl group, root system and root subgroups; 24. A BN-pair for GF; 25. Tori and Sylow subgroups; 26. Subgroups of maximal rank; 27. Maximal subgroups of finite classical groups; 28. About the classes CF1, …, CF7 and S; 29. Exceptional groups of Lie type; 30. Exercises for Part III; Appendix A. Root systems; Appendix B. Subsystems; Appendix C. Automorphisms of root systems; References; Index.

About the Author :
Gunter Malle is a Professor in the Department of Mathematics at the University of Kaiserslautern, Germany. Donna Testerman is a Lecturer in the Basic Sciences Faculty at the École Polytechnique Fédérale de Lausanne, Switzerland.

Review :
"This book provides a concise introduction to the theory of linear algebraic groups over an algebraically closed field (of arbitrary charachteristic) and the closely related finite groups of Lie type. Although there are several good books covering a similar range of topics, some important recent developments are treated here for the first time. This book is well written and the style of exposition is clear and reader-friendly, making it suitable for graduate students. The content is well organized, and the authors have sensibly avoided overloading the text with technical details." Timothy C. Burness for Mathematical Reviews