SOLUTION OF THE K(GV) PROBLEM (Icp Advanced Texts in 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. 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.