Characters and blocks of finite groups.(English)Zbl 0903.20004
London Mathematical Society Lecture Note Series. 250. Cambridge: Cambridge University Press. x, 287 p. (1998).
This book is the first to present the modular representation theory of finite groups from a character theoretic viewpoint. There are other approaches to this theory, e.g. using rings and modules. To get the full picture you may try to find a suitable balance between these approaches. This has been done in several previous books, e.g. byW. Feit [The representation theory of finite groups (1982;Zbl 0493.20007)]and byH. Nagao, Y. Tsushima [Representations of finite groups (1989;Zbl 0673.20002)]. Chosing a definite approach also has advantages as demonstrated by this book (and also by an earlier book ofJ. L. Alperin [Local representation theory (1986;Zbl 0593.20003)], using modules).
After an introductory chapter on algebras, Brauer characters are introduced in Chapter 2 and a number of fundamental properties are proved. Already here the book goes beyond other textbooks, e.g. by proving a result of Okuyama on groups, where all 2-modular character degrees are odd. Then follow chapters on blocks and the three main theorems in block theory. Each chapter contains nontrivial results which to the knowledge of the reviewer have not been in textbooks before like Robinson’s nice theorem on the number of blocks with a given defect group. Chapter 7 deals with Glauberman’s \(Z^*\)-theorem as an application of previous results. The next chapter considers in more detail induction and restriction of Brauer characters and Clifford theory. The final chapters deal with blocks in \(p\)-solvable groups and groups with a cyclic Sylow subgroup of order \(p\).
This is a nice and well written book. It assumes knowledge of ordinary character theory, but otherwise it more or less starts from scratch. It presents deep results in an easily accessible way and features the textbook debut of many interesting theorems. Thus it is also valuable as a place of reference. Its style seems to be influenced byI. M. Isaacs’ excellent book on ordinary character theory [Character theory of finite groups (1976;Zbl 0337.20005)]. In the introduction the author expresses his deep admiration for the work of Richard Brauer, the founder of modular representation theory. Brauer might have enjoyed this book, because the character theoretic approach was featured prominently in his research.
After an introductory chapter on algebras, Brauer characters are introduced in Chapter 2 and a number of fundamental properties are proved. Already here the book goes beyond other textbooks, e.g. by proving a result of Okuyama on groups, where all 2-modular character degrees are odd. Then follow chapters on blocks and the three main theorems in block theory. Each chapter contains nontrivial results which to the knowledge of the reviewer have not been in textbooks before like Robinson’s nice theorem on the number of blocks with a given defect group. Chapter 7 deals with Glauberman’s \(Z^*\)-theorem as an application of previous results. The next chapter considers in more detail induction and restriction of Brauer characters and Clifford theory. The final chapters deal with blocks in \(p\)-solvable groups and groups with a cyclic Sylow subgroup of order \(p\).
This is a nice and well written book. It assumes knowledge of ordinary character theory, but otherwise it more or less starts from scratch. It presents deep results in an easily accessible way and features the textbook debut of many interesting theorems. Thus it is also valuable as a place of reference. Its style seems to be influenced byI. M. Isaacs’ excellent book on ordinary character theory [Character theory of finite groups (1976;Zbl 0337.20005)]. In the introduction the author expresses his deep admiration for the work of Richard Brauer, the founder of modular representation theory. Brauer might have enjoyed this book, because the character theoretic approach was featured prominently in his research.
Reviewer: J.B.Olsson (København)
MSC:
| 20C20 | Modular representations and characters |
| 20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |
