LEADER 05268nam 2200661 a 450 001 9910813176703321 005 20240402140426.0 010 $a1-281-86946-5 010 $a9786611869465 010 $a1-86094-971-1 035 $a(CKB)1000000000400256 035 $a(EBL)1679367 035 $a(OCoLC)879023533 035 $a(SSID)ssj0000248488 035 $a(PQKBManifestationID)11235968 035 $a(PQKBTitleCode)TC0000248488 035 $a(PQKBWorkID)10202232 035 $a(PQKB)10342188 035 $a(MiAaPQ)EBC1679367 035 $a(WSP)0000P532 035 $a(Au-PeEL)EBL1679367 035 $a(CaPaEBR)ebr10255784 035 $a(CaONFJC)MIL186946 035 $a(PPN)16827678X 035 $a(EXLCZ)991000000000400256 100 $a20070920d2007 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 14$aThe solution of the k(GV) problem /$fPeter Schmid 205 $a1st ed. 210 $aLondon $cImperial College Press ;$aSingapore ;$aHackensack, NJ $cDistributed by World Scientific Pub.$dc2007 215 $a1 online resource (248 p.) 225 1 $aICP advanced texts in mathematics ;$vv. 4 300 $aDescription based upon print version of record. 311 $a1-86094-970-3 320 $aIncludes bibliographical references (p. 225-229) and index. 327 $aContents; Preface; 1. Conjugacy Classes, Characters, and Clifford Theory; 1.1 Class Functions and Characters; 1.2 Induced and Tensor-induced Modules; 1.3 Schur's Lemma; 1.4 Brauer's Permutation Lemma; 1.5 Algebraic Conjugacy; 1.6 Coprime Actions; 1.7 Invariant and Good Conjugacy Classes; 1.8 Nonstable Clifford Theory; 1.9 Stable Clifford Theory; 1.10 Good Conjugacy Classes and Extendible Characters; 2. Blocks of Characters and Brauer's k(B) Problem; 2.1 Modular Decomposition and Brauer Characters; 2.2 Cartan Invariants and Blocks; 2.3 Defect and Defect Groups; 2.4 The Brauer-Feit Theorem 327 $a2.5 Higher Decomposition Numbers, Subsections2.6 Blocks of p-Solvable Groups; 2.7 Coprime FpX-Modules; 3. The k(GV ) Problem; 3.1 Preliminaries; 3.2 Transitive Linear Groups; 3.3 Subsections and Point Stabilizers; 3.4 Abelian Point Stabilizers; 4. Symplectic and Orthogonal Modules; 4.1 Self-dual Modules; 4.2 Extraspecial Groups; 4.3 Holomorphs; 4.4 Good Conjugacy Classes Once Again; 4.5 Some Weil Characters; 4.6 Symplectic and Orthogonal Modules; 5. Real Vectors; 5.1 Regular, Abelian and Real Vectors; 5.2 The Robinson{Thompson Theorem; 5.3 Search for Real Vectors; 5.4 Clifford Reduction 327 $a5.5 Reduced Pairs5.6 Counting Methods; 5.7 Two Examples; 6. Reduced Pairs of Extraspecial Type; 6.1 Nonreal Reduced Pairs; 6.2 Fixed Point Ratios; 6.3 Point Stabilizers of Exponent 2; 6.4 Characteristic 2; 6.5 Extraspecial 3-Groups; 6.6 Extraspecial 2-Groups of Small Order; 6.7 The Remaining Cases; 7. Reduced Pairs of Quasisimple Type; 7.1 Nonreal Reduced Pairs; 7.2 Regular Orbits; 7.3 Covering Numbers, Projective Marks; 7.4 Sporadic Groups; 7.5 Alternating Groups; 7.6 Linear Groups; 7.7 Symplectic Groups; 7.8 Unitary Groups; 7.9 Orthogonal Groups; 7.10 Exceptional Groups 327 $a8. Modules without Real Vectors8.1 Some Fixed Point Ratios; 8.2 Tensor Induction of Reduced Pairs; 8.3 Tensor Products of Reduced Pairs; 8.4 The Riese-Schmid Theorem; 8.5 Nonreal Induced Pairs, Wreath Products; 9. Class Numbers of Permutation Groups; 9.1. The Partition Function; 9.2 Preparatory Results; 9.3 The Liebeck-Pyber Theorem; 9.4 Improvements; 10. The Final Stages of the Proof; 10.1 Class Numbers for Nonreal Reduced Pairs; 10.2 Counting Invariant Conjugacy Classes; 10.3 Nonreal Induced Pairs; 10.4 Characteristic 5; 10.5 Summary; 11. Possibilities for k(GV ) = jV j; 11.1 Preliminaries 327 $a11.2 Some Congruences11.3 Reduced Pairs; 12. Some Consequences for Block Theory; 12.1 Brauer Correspondence; 12.2 Clifford Theory of Blocks; 12.3 Blocks with Normal Defect Groups; 13. The Non-Coprime Situation; Appendix A: Cohomology of Finite Groups; Appendix B: Some Parabolic Subgroups; Appendix C: Weil Characters; Bibliography; List of Symbols; Index 330 $a The k(GV) conjecture claims that the number of conjugacy classes (irreducible characters) of the semidirect product GV is bounded above by the order of V. Here V is a finite vector space and G a subgroup of GL(V) of order prime to that of V. It may be regarded as the special case of Brauer's celebrated k(B) problem dealing with p-blocks B of p-solvable groups (p a prime). Whereas Brauer's problem is still open in its generality, the k(GV) problem has recently been solved, completing the work of a series of aut 410 0$aImperial College Press advanced texts in mathematics ;$vv. 4. 606 $aKernel functions 615 0$aKernel functions. 676 $a515/.7223 700 $aSchmid$b Peter$f1941-$01640147 712 02$aImperial College of Science, Technology and Medicine. 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910813176703321 996 $aThe solution of the k(GV) problem$93983552 997 $aUNINA