The subgroup structure of the finite classical groups.

*(English)*Zbl 0697.20004
London Mathematical Society Lecture Note Series, 129. Cambridge etc.: Cambridge University Press. x, 303 p. £17.50; $ 29.95 (1990).

The book is a substantial contribution to the study of the subgroups of the finite classical groups, that is, the linear, unitary, symplectic and orthogonal groups on finite vector spaces. For each classical group G, a class \({\mathcal C}(G)\) of natural (or ‘obvious’) subgroups of G is defined. In this book three problems are solved for these subgroups: (1) structure, (2) conjugacy, and (3) maximality.

The results are complicated and technical, and the statements alone occupy the whole of chapter 3 (pages 57-79). In part this complication arises from the authors’ laudable aim of leaving nothing undone.

Chapter 1 is a good general introduction to the maximal subgroup problem for the classical groups, and related problems. In conjunction with the extensive bibliography this should prove a useful short survey.

Chapter 2 describes the main properties of the classical groups that are needed in the rest of the book, and also serves to define the rather idiosyncratic notation. The authors’ hope that this chapter will stand on its own as an introduction to the classical groups seems however rather optimistic: it has the virtues of conciseness and compendiousness, which will be much appreciated by those who already know something of the subject, but which perhaps make it less suitable for the beginning graduate student.

Chapter 4 contains the proofs of all the results on structure and conjugacy, which leaves Chapters 5-8 for the much harder problems of maximality.

The results are complicated and technical, and the statements alone occupy the whole of chapter 3 (pages 57-79). In part this complication arises from the authors’ laudable aim of leaving nothing undone.

Chapter 1 is a good general introduction to the maximal subgroup problem for the classical groups, and related problems. In conjunction with the extensive bibliography this should prove a useful short survey.

Chapter 2 describes the main properties of the classical groups that are needed in the rest of the book, and also serves to define the rather idiosyncratic notation. The authors’ hope that this chapter will stand on its own as an introduction to the classical groups seems however rather optimistic: it has the virtues of conciseness and compendiousness, which will be much appreciated by those who already know something of the subject, but which perhaps make it less suitable for the beginning graduate student.

Chapter 4 contains the proofs of all the results on structure and conjugacy, which leaves Chapters 5-8 for the much harder problems of maximality.

Reviewer: R.A.Wilson

##### MSC:

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

20G40 | Linear algebraic groups over finite fields |

20E28 | Maximal subgroups |

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

20D30 | Series and lattices of subgroups |