有限群的特征论 Character Theory of Finite Groups

基本信息

Format Paperback | 336 pages

Dimensions 137 x 216 x 16mm | 344.73g

Publication date 02 Nov 2011

Publisher Dover Publications Inc.

Edition Statement Reprint

ISBN10 0486680142

ISBN13 9780486680149

信息仅供参考，具体以实物为准

书籍简介

这本书让人读起来很愉快。毫无疑问，它将成为，而且理应成为一本广泛使用的教科书和参考书。——《美国数学会公报》。

特征理论为证明有关有限群的定理提供了一个强有力的工具。除了处理将字符应用于纯群理论的技术外，本书的很大一部分内容都是关于字符本身的特性，以及这些特性如何反映和反映在群的结构中。

章节一由环论的初步知识组成。第二至第六章和第八章包含了字符理论的基本材料，第七章则处理了字符在群论中应用的重要技术。第9章考虑了任意场上的不可减代表，导致第10章对复数子场的关注。在第15章中，作者介绍了布劳尔的块和模块化字符理论。其余各章涉及更多的专业主题，如不可还原字符的度集与群的结构之间的联系。每一章后面都有一些经过仔细思考的问题，包括练习、例子、进一步的结果以及文本中定理的扩展和变化。

本书的先决条件是一些基本的有限群理论：Sylow定理、互换群的基本属性以及可解和无potent群。对环和伽罗瓦理论有一定的了解也很有用。简而言之，一年级研究生代数课程的内容应该是足够的准备。

The book is a pleasure to read. There is no question but that it will become, and deserves to be, a widely used textbook and reference. -- Bulletin of the American Mathematical Society.

Character theory provides a powerful tool for proving theorems about finite groups. In addition to dealing with techniques for applying characters to pure group theory, a large part of this book is devoted to the properties of the characters themselves and how these properties reflect and are reflected in the structure of the group.

Chapter I consists of ring theoretic preliminaries. Chapters 2 to 6 and 8 contain the basic material of character theory, while Chapter 7 treats an important technique for the application of characters to group theory. Chapter 9 considers irreducible representations over arbitrary fields, leading to a focus on subfields of the complex numbers in Chapter 10. In Chapter 15 the author introduces Brauer's theory of blocks and modular characters. Remaining chapters deal with more specialized topics, such as the connections between the set of degrees of the irreducible characters and structure of a group. Following each chapter is a selection of carefully thought out problems, including exercises, examples, further results and extensions and variations of theorems in the text.

Prerequisites for this book are some basic finite group theory: the Sylow theorems, elementary properties of permutation groups and solvable and nilpotent groups. Also useful would be some familiarity with rings and Galois theory. In short, the contents of a first-year graduate algebra course should be sufficient preparation.

作者简介

I. 马丁·艾萨克斯（Martin "Marty" Isaacs）是群论家和表示论家，威斯康星大学麦迪逊分校的荣誉数学教授。他目前住在加州伯克利，偶尔参加MathOverflow的活动。

I. Martin "Marty" Isaacs is a group theorist and representation theorist and professor emeritus of mathematics at the University of Wisconsin–Madison. He currently lives in Berkeley, California and is an occasional participant on MathOverflow.

目录

Preface; Acknowledgments; Notation

1. Algebras, modules, and representations. Problems

2. Group representations and characters. Problems

3. Characters and integrality. Problems

4. Products of characters. Problems

5. Induced characters. Problems

6. Normal subgroups. Problems

7. T.I. sets and exceptional characters. Problems

8. Brauer's theorem. Problems

9. Changing the field. Problems

10. The Schur index. Problems

11. Projective representations. Problems

12. Character degrees. Problems

13. Character correspondence. Problems

14. Linear groups. Problems

15. Changing the characteristic. Problems

Appendix. Some character tables

Biographic notes; References; Index