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.
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 Ecole Polytechnique Federale 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.