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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.
Sample Chapter(s)
Chapter 1: Conjugacy Classes, Characters, and Clifford Theory (296 KB)
Contents:
- Conjugacy Classes, Characters and Clifford Theory
- Blocks of Characters and Brauer's k(B) Problem
- The k(GV) Problem
- Symplectic and Orthogonal Modules
- Real Vectors
- Reduced Pairs of Extraspecial Type
- Reduced Pairs of Quasisimple Type
- Modules Without Real Vectors
- Class Numbers of Permutation Groups
- The Final Stages of the Proof
- Possibilities for k(GV) = |V|
- Some Consequences for Block Theory
- The Non-Coprime Situation
Readership: Postgraduate students and researchers with background and research interests in group and representation theory.
