Solution of the K(GV) Problem

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...
More Description

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 TheoryBlocks of Characters and Brauer's "k(B)" ProblemThe "k(GV)" ProblemSymplectic and Orthogonal ModulesReal VectorsReduced Pairs of Extraspecial TypeReduced Pairs of Quasisimple TypeModules Without Real VectorsClass Numbers of Permutation GroupsThe Final Stages of the ProofPossibilities for "k(GV)" = |"V"|Some Consequences for Block TheoryThe Non-Coprime Situation
Readership: Postgraduate students and researchers with background and research interests in group and representation theory.