Solution Of The K(gv) Problem, The

Peter Schmid

• Publisher: ICP
• Year: 2007
• Print ISBN:
  9781860949708
  9781860949708
• eBook ISBN:
  9781860949715
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

The k(GV) conjecture claims that the number of conjugacy classes (irreducible characters) of the semidirect product GV is bounded above by the order of V. Here V is a finite vector space and G a subgroup of GL(V) of order prime to that of V. It may be regarded as the special case of Brauer's celebrated k(B) problem dealing with p-blocks B of p-solvable groups (p a prime). Whereas Brauer's problem is still open in its generality, the k(GV) problem has recently been solved, completing the work of a series of authors over a period of more than forty years. In this book the developments, ideas and methods, leading to this remarkable result, are described in detail.

Table of contents

• Contents
• Preface
• 1. Conjugacy Classes, Characters, and Clifford Theory
• 2. Blocks of Characters and Brauer's k(B) Problem
• 3. The k(GV ) Problem
• 4. Symplectic and Orthogonal Modules
• 5. Real Vectors
• 6. Reduced Pairs of Extraspecial Type
• 7. Reduced Pairs of Quasisimple Type
• 8. Modules without Real Vectors
• 9. Class Numbers of Permutation Groups
• 10. The Final Stages of the Proof
• 11. Possibilities for k(GV ) = jV j
• 12. Some Consequences for Block Theory
• 13. The Non-Coprime Situation
• Appendix A: Cohomology of Finite Groups
• Appendix B: Some Parabolic Subgroups
• Appendix C: Weil Characters
• Bibliography
• List of Symbols
• Index