The solution of the k(GV) problem [[electronic resource] /] / Peter Schmid

• Autore: Schmid Peter <1941->
• Pubblicazione: London, : Imperial College Press; Singapore ; ; Hackensack, NJ, : Distributed by World Scientific Pub., c2007
• Descrizione fisica: 1 online resource (248 p.)
• Disciplina: 515/.7223
• Soggetto topico: Kernel functions
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. 225-229) and index.
• Nota di contenuto: Contents; 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 ; 2.5 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 ; 5.5 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 ; 8. 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 ; 11.2 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
• Sommario/riassunto: The <i>k(GV)</i> conjecture claims that the number of conjugacy classes (irreducible characters) of the semidirect product <i>GV</i> is bounded above by the order of <i>V</i>. Here <i>V</i> is a finite vector space and <i>G</i> a subgroup of <i>GL(V)</i> of order prime to that of <i>V</i>. It may be regarded as the special case of Brauer's celebrated <i>k(B)</i> problem dealing with <i>p</i>-blocks <i>B</i> of p-solvable groups (<i>p</i> a prime). Whereas Brauer's problem is still open in its generality, the <i>k(GV)</i> problem has recently been solved, completing the work of a series of aut
• Titolo autorizzato: The solution of the k(GV) problem
• ISBN: 1-281-86946-5; 9786611869465; 1-86094-971-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910784795003321
• Lo trovi qui: Univ. Federico II
• Serie: Imperial College Press advanced texts in mathematics ; ; v. 4.