Linear Algebraic Groups and Finite Groups of Lie Type

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.

• 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.

"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