Character Theory of Finite Groups
About this Title
I. Martin Isaacs, University of Wisconsin, Madison, Madison, WI
Publication: AMS Chelsea Publishing
Publication Year: 1976; Volume 359
ISBNs: 978-0-8218-4229-4 (print); 978-1-4704-3035-1 (online)
Character theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group.
The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging.
In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters.
This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory.
Graduate students and research mathematicians interested in finite groups, character theory, and representation theory.
Table of Contents
- Chapter 1. Algebras, modules, and representations
- Chapter 2. Group representations and characters
- Chapter 3. Characters and integrality
- Chapter 4. Products of characters
- Chapter 5. Induced characters
- Chapter 6. Normal subgroups
- Chapter 7. T.I. sets and exceptional characters
- Chapter 8. Brauer’s theorem
- Chapter 9. Changing the field
- Chapter 10. The Schur index
- Chapter 11. Projective representations
- Chapter 12. Character degrees
- Chapter 13. Character correspondence
- Chapter 14. Linear groups
- Chapter 15. Changing the characteristic
- Appendix. Some character tables