Linear Algebraic Groups and Finite Groups of Lie Type

The first textbook on the subgroup structure, in particular maximal subgroups, for both algebraic and finite groups of Lie type.

Gunter Malle (Author), Donna Testerman (Author)

9781107008540, Cambridge University Press

Hardback, published 8 September 2011

324 pages, 6 b/w illus. 20 tables 100 exercises
23.1 x 15.5 x 2 cm, 0.59 kg

"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

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.

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.

Subject Areas: Topology [PBP], Algebra [PBF], Mathematics [PB]