The 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 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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2-group absolutely irreducible aﬀorded assume Atlas Aut(L automorphism block Brauer character CG(v character table conjugacy classes conjugate contains coprime cosets counting argument Cq(v cyclic of order decomposition defect group diagonal dimension divisor eigenspace eigenvalues elements of order extending extraspecial type faithful irreducible characters FG-module ﬁeld ﬁnite follows g G G G-orbit GL(V Hence induced integer involution Irr(B Irr(X irreducible characters irreducible constituent isomorphic k(GV kg(NV Lemma Let g linear character maximal subgroup modp nonabelian noncentral element nonreal reduced pairs nontrivial nonzero vectors normal subgroup orthogonal otherwise p-block pair G,V permutation group point stabilizers prime order Proof quasisimple quotient group rational-valued real for G regular orbit regular vector roots of unity Schur Schur index self-dual simple groups Singer cycle standard holomorph standard module strongly real vector subgroup of G subspace Sylow p-subgroup symplectic Theorem 4.4 Theorem 7.2a unique wreath product