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Linear Algebraic Groups and Finite Groups of Lie Type

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Part of Cambridge Studies in Advanced Mathematics

  • Date Published: October 2011
  • availability: Available
  • format: Hardback
  • isbn: 9781107008540

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About the Authors
  • 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.

    • Tried and tested for graduate courses
    • Includes numerous exercises and examples ranging in difficulty
    • The first textbook to present the modern results on maximal subgroups of both the semisimple algebraic groups and the finite groups of Lie type
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    Reviews & endorsements

    "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

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    Product details

    • Date Published: October 2011
    • format: Hardback
    • isbn: 9781107008540
    • length: 324 pages
    • dimensions: 231 x 155 x 20 mm
    • weight: 0.59kg
    • contains: 6 b/w illus. 20 tables 100 exercises
    • availability: Available
  • Table of Contents

    List of tables
    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

  • Resources for

    Linear Algebraic Groups and Finite Groups of Lie Type

    Gunter Malle, Donna Testerman

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  • Authors

    Gunter Malle, Technische Universität Kaiserslautern, Germany
    Gunter Malle is a Professor in the Department of Mathematics at the University of Kaiserslautern, Germany.

    Donna Testerman, École Polytechnique Fédérale de Lausanne
    Donna Testerman is a Lecturer in the Basic Sciences Faculty at the École Polytechnique Fédérale de Lausanne, Switzerland.

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