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.

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