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Lectures on Profinite Topics in Group Theory.
An introduction to three key aspects of current research in infinite group theory, suitable for graduate students.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
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| Colección: | London Mathematical Society Student Texts, 77.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Contents; Preface; Editor's introduction; I An introduction to compact p-adic Lie groups; 1 Introduction; 2 From finite p-groups to compact p-adic Lie groups; 2.1 Nilpotent groups; 2.2 Finite p-groups; 2.3 Lie rings; 2.4 Applying Lie methods to groups; 2.5 Absolute values; 2.6 p-adic numbers; 2.7 p-adic integers; 2.8 Preview: p-adic analytic pro-p groups; 3 Basic notions and facts from point-set topology; 4 First series of exercises; 5 Powerful groups, profinite groups and pro-p groups; 5.1 Powerful finite p-groups; 5.2 Profinite groups as Galois groups.
- 5.3 Profinite groups as inverse limits5.4 Profinite groups as profinite completions; 5.5 Profinite groups as topological groups; 5.6 Pro-p groups; 5.7 Powerful pro-p groups; 5.8 Pro-p groups of finite rank
- summary ofcharacterisations; 6 Second series of exercises; 7 Uniformly powerful pro-p groups and Zp-Lie lattices; 7.1 Uniformly powerful pro-p groups; 7.2 Associated additive structure; 7.3 Associated Lie structure; 7.4 The Hausdorff formula; 7.5 Applying the Hausdorff formula; 8 The group GLd(Zp), just-infinite pro-p groups and the Lie correspondence for saturable pro-p groups.
- 8.1 The group GLd(Zp)
- an example8.2 Just-infinite pro-p groups; 8.3 Potent filtrations and saturable pro-p groups; 8.4 Lie correspondence; 9 Third series of exercises; 10 Representations of compact p-adic Lie groups; 10.1 Representation growth and Kirillov's orbit method; 10.2 The orbit method for saturable pro-p groups; 10.3 An application of the orbit method; References for Chapter I; II Strong approximation methods; 1 Introduction; 2 Algebraic groups; 2.1 The Zariski topology on Kn; 2.2 Linear algebraic groups as closed subgroups of GLn(K); Basic examples.
- Basic properties of Algebraic groupsFields of definition and restriction of scalars; The Lie algebra of G; Connection with Lie algebras of locally compact topological groups; 2.3 Semisimple algebraic groups: the classification ofsimply connected algebraic groups over K; 2.4 Reductive groups; 2.5 Chevalley groups; 3 Arithmetic groups and the congruence topology; 3.1 Rings of algebraic integers in number fields; 3.2 The congruence topology on GLn(k) and GLn(O); Valuations of k; 3.3 Arithmetic groups; 4 The strong approximation theorem; 4.1 An aside: Serre's conjecture; 5 Lubotzky's alternative.
- 6 Applications of Lubotzky's alternative6.1 The finite simple groups of Lie type; 6.2 Refinements; 6.3 Normal subgroups of linear groups; 6.4 Representations, sieves and expanders; 7 The Nori
- Weisfeiler theorem; 7.1 Unipotently generated subgroups of algebraic groups over finite fields; 8 Exercises; References for Chapter II; III A newcomer's guideto zeta functions of groups and rings; 1 Introduction; 1.1 Zeta functions of groups; 1.2 Zeta functions of rings; 1.3 Linearisation; 1.4 Organisation of the chapter; 2 Local and global zeta functions; 2.1 Rationality and variation with the prime.