The Solution of the K(GV) Problem

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Imperial College Press, 2007 - Mathematics - 248 pages
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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.
 

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