Finite group theory.

*(English)*Zbl 0583.20001
Cambridge Studies in Advanced Mathematics, 10. Cambridge etc.: Cambridge University Press. IX, 274 p. £22.50; $ 32.50 (1986).

Nowadays there is a number of fine books on finite group theory available: for instance ”Gorenstein; Finite groups”, ”Huppert (Blackburn); Endliche Gruppen (Finite groups) I-III”, and ”Suzuki, Group theory I-II”. So what is the motivation for one of the leading group theorists to write a new book on this subject?

Chapter 1 introduces in full generality the central notions of this book. First it lists without proofs the elementary properties of groups. Then the language of categories, of graphs and geometries (in the sense of Tits) is explained. Furthermore the general concept of a group representation is given.

Chapter 2 deals with basic properties of permutation representations and covers Sylow’s theorem and related topics.

In chapter 3 the author investigates representations in the sense, that one group acts on another. In this section fall the treatment of normal series, the Jordan-Hölder theorem, and characteristic subgroups. The basic properties of solvable and nilpotent groups are developed. Finally semidirect products (in particular central products and wreath products) are investigated. This reaches as far as to Gaschütz’ criterion on the splitting of group extensions.

Chapter 4 gives an introduction to linear groups. Besides basics as Maschke’s and Clifford’s theorems and semisimplicity it treats the groups GL(V) and SL(V). A not so standard topic: the dual representation is studied with great care.

Chapter 5 gives besides the fundamental concepts of permutation groups a closer look on the groups \(A_ n\) and \(S_ n\). Again not in the norm: quite thoroughly the rank 3-permutation groups are explored.

Chapter 6 is on extensions of groups and modules. The 1-cohomology is discussed and its connections with group complements and extensions of group modules by the trivial module. Here we find also the Schur- Zassenhaus theorem and Hall’s generalization of Sylow’s theorem. Finally, Aschbacher considers the action of a group on others of coprime order.

In chapter 7 finite vector spaces with nondegenerate sesquilinear forms (or quadratic forms) and the isometry groups are treated extensively. This also includes a very short proof of Witt’s theorem. However more subtile concepts as generation of classical groups, the Clifford algebra, and the spinor norm are subject of this section too.

p-groups are the theme of chapter 8. Starting with basic concepts the author pushes forward to a closer investigation of extremal p-groups, i.e. nonabelian p-groups with cyclic subgroups of index p and groups of symplectic type. Another main point: the action of p’-groups on p-groups (critical subgroups, Thompson’s A B-lemma etc.). This part resembles closely to the analogous part in Gorenstein’s book.

In chapter 9 Aschbacher studies how linear representations behave under field extensions. The action of the Galois group on the representation tensored by an extension field is considered, splitting fields are introduced. Special care is given to finite fields. We find a thorough discussion of the field of definition of a representation - again a subject outside of the usual frame. Finally consequences from the concept of the minimal polynomial - like the Jordan-Chevalley decomposition - are drawn.

Presentations of groups is the title of chapter 10. After the basics the author considers with great completeness Coxeter groups and their representations as reflection groups on euclidean spaces. The last section of this chapter introduces root systems and the action of the Weyl group on a root system.

In chapter 11 components and the generalized Fitting group are defined. One also finds here first preparations for the signalizer functor theorem. The second item: the Thompson factorization and the Thompson group. This chapter ends with a discussion of central extensions, the universal covering group, and the Schur multiplier. In particular the Schur multiplier of \(A_ n\) is computed.

In chapter 12 one finds the basic representation theory in characteristic 0. The first part is fairly standard and ends with Burnside’s \(p^ aq^ b\)-theorem and the existence of the Frobenius kernel. However the last part of this chapter contains special results on representations of solvable groups. In part they are used for the signalizer functor theorem.

Chapter 13 is named transfer and fusion. After an introduction to transfer Alperin’s fusion theorem is proved. Then results on p- complements (Burnside, Frobenius, Thompson) and the Baer-Suzuki theorem are discussed. As a highpoint the author finally proves the nilpotency of the Frobenius kernel.

Chapter 14 provides an introduction to the geometry of groups (in the sense of Tits). It starts with very general concepts as complexes and chambers, and specializes then to Coxeter complexes, buildings, and (B,N)-pairs. The basic theory is developed quite far. As an application Aschbacher proves the simplicity of the classical groups.

Chapter 15 is devoted to a new proof (using ideas of Bender) of the (solvable) signalizer functor theorem.

Chapter 16 deals with finite simple groups and their classification. It begins with a look at involutions in finite groups, i.e. the Brauer- Fowler theorem and the Thompson order formula. Then Aschbacher gives the notion of connected groups and studies techniques which come up with this concept. Next the author studies the simple groups of Lie type in more detail. Among others he proves in essence the Borel-Tits theorem. This section ends with an outline of the proof of the classification theorem.

As I tried already to indicate, this book has a distinct own flavor. Some large parts fall outside the content of usual text books. Also the more standard results are varied in different directions to obtain highest generality. Main object of this book is to explain the principal methods of the working group theorist. In contrast to other books Aschbacher will usually not present complete structure theory (of some part of group theory). The tools the author provides have main applications for finite simple groups. Another goal of this book is to pave the way for the geometric treatment of finite groups via chamber systems.

The proofs in this book are quite short and demanding. Proofs of easier results are skipped or left as exercises. Every chapter has exercises which are equally demanding. Often these exercises are little theorems of own importance. Definitions are usually not explained by examples. However every chapter starts and ends with some comments, which motivate results, outline connections with other parts of the book and indicate further developments. The short list of references contains only a few books and articles of greater importance. The misprints I noticed can be easily corrected from the context.

Conclusion: Aschbacher has written a book with a marked personal character. Starting from the fundamental notion of group representation this book develops the important methods in group theory. This book resembles Gorenstein’s monograph but also reflects the further developments since 1967. The amount of material covered by some 260 pages is impressive. To read this book without help might be tough for a beginner - in particular with no prerequisites in this field. However for any one who works seriously with finite groups - in particular does research - this excellent book is a most useful source and reference. It also provides a true picture in which spirit group theory is done today.

Chapter 1 introduces in full generality the central notions of this book. First it lists without proofs the elementary properties of groups. Then the language of categories, of graphs and geometries (in the sense of Tits) is explained. Furthermore the general concept of a group representation is given.

Chapter 2 deals with basic properties of permutation representations and covers Sylow’s theorem and related topics.

In chapter 3 the author investigates representations in the sense, that one group acts on another. In this section fall the treatment of normal series, the Jordan-Hölder theorem, and characteristic subgroups. The basic properties of solvable and nilpotent groups are developed. Finally semidirect products (in particular central products and wreath products) are investigated. This reaches as far as to Gaschütz’ criterion on the splitting of group extensions.

Chapter 4 gives an introduction to linear groups. Besides basics as Maschke’s and Clifford’s theorems and semisimplicity it treats the groups GL(V) and SL(V). A not so standard topic: the dual representation is studied with great care.

Chapter 5 gives besides the fundamental concepts of permutation groups a closer look on the groups \(A_ n\) and \(S_ n\). Again not in the norm: quite thoroughly the rank 3-permutation groups are explored.

Chapter 6 is on extensions of groups and modules. The 1-cohomology is discussed and its connections with group complements and extensions of group modules by the trivial module. Here we find also the Schur- Zassenhaus theorem and Hall’s generalization of Sylow’s theorem. Finally, Aschbacher considers the action of a group on others of coprime order.

In chapter 7 finite vector spaces with nondegenerate sesquilinear forms (or quadratic forms) and the isometry groups are treated extensively. This also includes a very short proof of Witt’s theorem. However more subtile concepts as generation of classical groups, the Clifford algebra, and the spinor norm are subject of this section too.

p-groups are the theme of chapter 8. Starting with basic concepts the author pushes forward to a closer investigation of extremal p-groups, i.e. nonabelian p-groups with cyclic subgroups of index p and groups of symplectic type. Another main point: the action of p’-groups on p-groups (critical subgroups, Thompson’s A B-lemma etc.). This part resembles closely to the analogous part in Gorenstein’s book.

In chapter 9 Aschbacher studies how linear representations behave under field extensions. The action of the Galois group on the representation tensored by an extension field is considered, splitting fields are introduced. Special care is given to finite fields. We find a thorough discussion of the field of definition of a representation - again a subject outside of the usual frame. Finally consequences from the concept of the minimal polynomial - like the Jordan-Chevalley decomposition - are drawn.

Presentations of groups is the title of chapter 10. After the basics the author considers with great completeness Coxeter groups and their representations as reflection groups on euclidean spaces. The last section of this chapter introduces root systems and the action of the Weyl group on a root system.

In chapter 11 components and the generalized Fitting group are defined. One also finds here first preparations for the signalizer functor theorem. The second item: the Thompson factorization and the Thompson group. This chapter ends with a discussion of central extensions, the universal covering group, and the Schur multiplier. In particular the Schur multiplier of \(A_ n\) is computed.

In chapter 12 one finds the basic representation theory in characteristic 0. The first part is fairly standard and ends with Burnside’s \(p^ aq^ b\)-theorem and the existence of the Frobenius kernel. However the last part of this chapter contains special results on representations of solvable groups. In part they are used for the signalizer functor theorem.

Chapter 13 is named transfer and fusion. After an introduction to transfer Alperin’s fusion theorem is proved. Then results on p- complements (Burnside, Frobenius, Thompson) and the Baer-Suzuki theorem are discussed. As a highpoint the author finally proves the nilpotency of the Frobenius kernel.

Chapter 14 provides an introduction to the geometry of groups (in the sense of Tits). It starts with very general concepts as complexes and chambers, and specializes then to Coxeter complexes, buildings, and (B,N)-pairs. The basic theory is developed quite far. As an application Aschbacher proves the simplicity of the classical groups.

Chapter 15 is devoted to a new proof (using ideas of Bender) of the (solvable) signalizer functor theorem.

Chapter 16 deals with finite simple groups and their classification. It begins with a look at involutions in finite groups, i.e. the Brauer- Fowler theorem and the Thompson order formula. Then Aschbacher gives the notion of connected groups and studies techniques which come up with this concept. Next the author studies the simple groups of Lie type in more detail. Among others he proves in essence the Borel-Tits theorem. This section ends with an outline of the proof of the classification theorem.

As I tried already to indicate, this book has a distinct own flavor. Some large parts fall outside the content of usual text books. Also the more standard results are varied in different directions to obtain highest generality. Main object of this book is to explain the principal methods of the working group theorist. In contrast to other books Aschbacher will usually not present complete structure theory (of some part of group theory). The tools the author provides have main applications for finite simple groups. Another goal of this book is to pave the way for the geometric treatment of finite groups via chamber systems.

The proofs in this book are quite short and demanding. Proofs of easier results are skipped or left as exercises. Every chapter has exercises which are equally demanding. Often these exercises are little theorems of own importance. Definitions are usually not explained by examples. However every chapter starts and ends with some comments, which motivate results, outline connections with other parts of the book and indicate further developments. The short list of references contains only a few books and articles of greater importance. The misprints I noticed can be easily corrected from the context.

Conclusion: Aschbacher has written a book with a marked personal character. Starting from the fundamental notion of group representation this book develops the important methods in group theory. This book resembles Gorenstein’s monograph but also reflects the further developments since 1967. The amount of material covered by some 260 pages is impressive. To read this book without help might be tough for a beginner - in particular with no prerequisites in this field. However for any one who works seriously with finite groups - in particular does research - this excellent book is a most useful source and reference. It also provides a true picture in which spirit group theory is done today.

Reviewer: U.Dempwolff

##### MSC:

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

20Dxx | Abstract finite groups |

20C15 | Ordinary representations and characters |

20F05 | Generators, relations, and presentations of groups |

20F65 | Geometric group theory |

20B30 | Symmetric groups |