Linear algebraic groups and finite groups of Lie type.

*(English)*Zbl 1256.20045
Cambridge Studies in Advanced Mathematics 133. Cambridge: Cambridge University Press (ISBN 978-1-107-00854-0/hbk). xiv, 309 p. (2011).

Originating from a summer school taught by the authors, this book provides a concise introduction to the theory of linear algebraic groups over an algebraically closed field (of arbitrary characteristic) 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.

The material is organized into three parts. In Part I, the fundamental results on linear algebraic groups over an algebraically closed field are developed. Necessarily, the content in Part I is not completely self-contained. For example, some familiarity with the basic notions of affine and projective varieties is assumed, together with some background results in commutative algebra and algebraic geometry. The last three chapters of this part are devoted to introducing the combinatorial data which classify semisimple algebraic groups and to establishing structural results and the classification theorem.

Part II develops the basic theory of the semisimple algebraic groups classified in Part I. Familiar topics covered here include the BN-pair structure, the Levi decomposition of parabolic subgroups, subsystem subgroups, automorphisms and the conjugacy classes and centralizers of semisimple elements. This material is largely based on the pioneering work of Borel, Bruhat, Springer, Steinberg and Tits in the 20-30 year period following Chevalley’s classification theorem in the mid-1950s. Later chapters introduce some more recent developments. In the last chapters of Part II the authors turn to more recent developments in the theory of semisimple algebraic groups, where they describe the classification of their maximal positive-dimensional subgroups. These results can be seen as an extension of the fundamental work of Dynkin on the maximal subalgebras of the semisimple complex Lie algebras. The final chapter of Part II provides a brief survey of the major project to classify the maximal positive-dimensional subgroups of the exceptional algebraic groups, which was completed in 2004 [M. W. Liebeck and G. M. Seitz, Mem. Am. Math. Soc. 802 (2004; Zbl 1058.20040)].

In Part III the authors introduce the finite analogues of the semisimple algebraic groups: the finite groups of Lie type. There are several standard texts on this topic (for example, R. W. Carter’s book [Simple groups of Lie type. Pure and Applied Mathematics. Vol. XXVIII. London etc.: John Wiley & Sons (1972; Zbl 0248.20015)]), but once again the treatment here has some unique features. Starting from the material introduced in Parts I and II, the authors define a finite group of Lie type as the set of fixed points of a suitable (i.e., non-algebraic) automorphism of a semisimple algebraic group over an algebraically closed field of prime characteristic. This modern viewpoint, originally due to Steinberg, provides a uniform approach to the finite groups of Lie type and allows methods and results from the rich theory of algebraic groups to be applied. One of the first results introduced in Part III is the Lang-Steinberg theorem. This is a basic result which underlies much of the interplay between the finite and algebraic groups. The classification of the Steinberg endomorphisms of a simple algebraic group is stated, and this yields a classification of the finite groups of Lie type. With this in hand, the authors introduce the root subgroups of a finite group of Lie type (corresponding to the root system of the ambient algebraic group), and an analogue of the Bruhat decomposition is obtained, which reveals the BN-pair structure of these finite groups, and much more. Once again, several carefully selected examples are used to illustrate the general theory. The remaining chapters in Part III deal with various aspects of the subgroup structure of finite groups of Lie type, building on the earlier analysis in Part II for semisimple algebraic groups. Familiar topics discussed here include maximal tori, Sylow subgroups, parabolic subgroups and Levi subgroups. The final three chapters provide an up-to-date survey on the maximal subgroups of finite groups of Lie type. Part III concludes with a brief survey of results on the maximal subgroups of finite exceptional groups, and a short appendix on root systems (including root subsystems and automorphisms) is given.

This book is well written and the style of presentation is clear and easy to read, which make it suitable for graduate students. The content is well organized, and the authors have sensibly avoided overloading the text with technical details (however, full references are provided throughout). Well-chosen examples are often used to describe the general theory, and the main text is complemented by many exercises of varying difficulty. This book will be particularly useful to a reader who wants to approach the study of the subgroup structure of simple groups, both finite and algebraic.

The material is organized into three parts. In Part I, the fundamental results on linear algebraic groups over an algebraically closed field are developed. Necessarily, the content in Part I is not completely self-contained. For example, some familiarity with the basic notions of affine and projective varieties is assumed, together with some background results in commutative algebra and algebraic geometry. The last three chapters of this part are devoted to introducing the combinatorial data which classify semisimple algebraic groups and to establishing structural results and the classification theorem.

Part II develops the basic theory of the semisimple algebraic groups classified in Part I. Familiar topics covered here include the BN-pair structure, the Levi decomposition of parabolic subgroups, subsystem subgroups, automorphisms and the conjugacy classes and centralizers of semisimple elements. This material is largely based on the pioneering work of Borel, Bruhat, Springer, Steinberg and Tits in the 20-30 year period following Chevalley’s classification theorem in the mid-1950s. Later chapters introduce some more recent developments. In the last chapters of Part II the authors turn to more recent developments in the theory of semisimple algebraic groups, where they describe the classification of their maximal positive-dimensional subgroups. These results can be seen as an extension of the fundamental work of Dynkin on the maximal subalgebras of the semisimple complex Lie algebras. The final chapter of Part II provides a brief survey of the major project to classify the maximal positive-dimensional subgroups of the exceptional algebraic groups, which was completed in 2004 [M. W. Liebeck and G. M. Seitz, Mem. Am. Math. Soc. 802 (2004; Zbl 1058.20040)].

In Part III the authors introduce the finite analogues of the semisimple algebraic groups: the finite groups of Lie type. There are several standard texts on this topic (for example, R. W. Carter’s book [Simple groups of Lie type. Pure and Applied Mathematics. Vol. XXVIII. London etc.: John Wiley & Sons (1972; Zbl 0248.20015)]), but once again the treatment here has some unique features. Starting from the material introduced in Parts I and II, the authors define a finite group of Lie type as the set of fixed points of a suitable (i.e., non-algebraic) automorphism of a semisimple algebraic group over an algebraically closed field of prime characteristic. This modern viewpoint, originally due to Steinberg, provides a uniform approach to the finite groups of Lie type and allows methods and results from the rich theory of algebraic groups to be applied. One of the first results introduced in Part III is the Lang-Steinberg theorem. This is a basic result which underlies much of the interplay between the finite and algebraic groups. The classification of the Steinberg endomorphisms of a simple algebraic group is stated, and this yields a classification of the finite groups of Lie type. With this in hand, the authors introduce the root subgroups of a finite group of Lie type (corresponding to the root system of the ambient algebraic group), and an analogue of the Bruhat decomposition is obtained, which reveals the BN-pair structure of these finite groups, and much more. Once again, several carefully selected examples are used to illustrate the general theory. The remaining chapters in Part III deal with various aspects of the subgroup structure of finite groups of Lie type, building on the earlier analysis in Part II for semisimple algebraic groups. Familiar topics discussed here include maximal tori, Sylow subgroups, parabolic subgroups and Levi subgroups. The final three chapters provide an up-to-date survey on the maximal subgroups of finite groups of Lie type. Part III concludes with a brief survey of results on the maximal subgroups of finite exceptional groups, and a short appendix on root systems (including root subsystems and automorphisms) is given.

This book is well written and the style of presentation is clear and easy to read, which make it suitable for graduate students. The content is well organized, and the authors have sensibly avoided overloading the text with technical details (however, full references are provided throughout). Well-chosen examples are often used to describe the general theory, and the main text is complemented by many exercises of varying difficulty. This book will be particularly useful to a reader who wants to approach the study of the subgroup structure of simple groups, both finite and algebraic.

Reviewer: Fiorenza Morini (Parma)

##### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

20D06 | Simple groups: alternating groups and groups of Lie type |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20G40 | Linear algebraic groups over finite fields |

20G05 | Representation theory for linear algebraic groups |