Representations of finite groups. Transl. from the Japanese.

*(English)*Zbl 0673.20002
Boston, MA etc.: Academic Press, Inc. xvii, 424 p. $ 70.00 (1989).

The origin of this book goes back to the last half of the 1950’s when one of the authors, Nagao, had planned to write a book on representation theory with T. Nakayama. However the plan was suspended because R. Brauer, the founder of modular representation theory, had told Nakayama that he planned to write a similar book. [Unfortunately Brauer never wrote his book, although some very interesting notes, based on his lectures in Japan 1959, were published in 1979 by the Kyoto University (Zbl 0574.20007).] Now Nagao and Tsushima, one of Nakayama’s students, have realized the project.

The book is a selfcontained and comprehensive introduction to important parts of ordinary and modular representation theory. It has 5 chapters.

Chapter 1 summarizes fundamental results on rings and modules needed in the subsequent chapters. Chapter 2 covers basic facts about algebras and their representations, simple and separable algebras, Frobenius and symmetric algebras, the Schur index and crossed products. Chapter 3 establishes the basics of the theory of ordinary and modular representations of finite groups, Clifford theory, projective representations and Brauer’s characterization of characters. Chapter 4 is concerned with the theory of indecomposable modules, vertices, sources, and Green correspondence, the Burry-Carlson theorem, Green’s indecomposability theorem with applications and Scott modules. Chapter 5 is devoted to block theory. The topics include the Brauer correspondence, Brauer’s 3 main theorems on blocks, Clifford theory of blocks, blocks of factor groups, the Alperin-BrouĂ© language of subpairs, lower defect groups and the Glauberman correspondence.

As it may be seen from this description most of the standard topics are covered as well as a few recent developments. Some important new aspects, like the Auslander-Reiten theory of almost split sequences and the use of cohomological methods (Carlson et at.), are missing. Also a definite lack of illuminating examples and applications is noticeable, even in the many problems and exercises. However the book is very carefully written with a clear exposition. It is well suited as a book of reference or (supplemented by examples) as a textbook for a course on representation theory. As already mentioned it contains many exercises and solved problems, which provide valuable additions to the results proved in the text.

The book is a selfcontained and comprehensive introduction to important parts of ordinary and modular representation theory. It has 5 chapters.

Chapter 1 summarizes fundamental results on rings and modules needed in the subsequent chapters. Chapter 2 covers basic facts about algebras and their representations, simple and separable algebras, Frobenius and symmetric algebras, the Schur index and crossed products. Chapter 3 establishes the basics of the theory of ordinary and modular representations of finite groups, Clifford theory, projective representations and Brauer’s characterization of characters. Chapter 4 is concerned with the theory of indecomposable modules, vertices, sources, and Green correspondence, the Burry-Carlson theorem, Green’s indecomposability theorem with applications and Scott modules. Chapter 5 is devoted to block theory. The topics include the Brauer correspondence, Brauer’s 3 main theorems on blocks, Clifford theory of blocks, blocks of factor groups, the Alperin-BrouĂ© language of subpairs, lower defect groups and the Glauberman correspondence.

As it may be seen from this description most of the standard topics are covered as well as a few recent developments. Some important new aspects, like the Auslander-Reiten theory of almost split sequences and the use of cohomological methods (Carlson et at.), are missing. Also a definite lack of illuminating examples and applications is noticeable, even in the many problems and exercises. However the book is very carefully written with a clear exposition. It is well suited as a book of reference or (supplemented by examples) as a textbook for a course on representation theory. As already mentioned it contains many exercises and solved problems, which provide valuable additions to the results proved in the text.

Reviewer: J.B.Olsson

##### MSC:

20Cxx | Representation theory of groups |

20C15 | Ordinary representations and characters |

20C20 | Modular representations and characters |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |