Format:
Online-Ressource (XIII, 371 S.)
Edition:
Online-Ausg. 2013 Electronic reproduction; Available via World Wide Web
ISBN:
9788847024212
Series Statement:
Unitext 58
Uniform Title:
Gruppi 〈engl.〉
Content:
Antonio Machì
Content:
Groups are a means of classification, via the group action on a set, but also the object of a classification. How many groups of a given type are there, and how can they be described? Hölder's program for attacking this problem in the case of finite groups is a sort of leitmotiv throughout the text. Infinite groups are also considered, with particular attention to logical and decision problems. Abelian, nilpotent and solvable groups are studied both in the finite and infinite case. Permutation groups and are treated in detail; their relationship with Galois theory is often taken into account. The last two chapters deal with the representation theory of finite group and the cohomology theory of groups; the latter with special emphasis on the extension problem. The sections are followed by exercises; hints to the solution are given, and for most of them a complete solution is provided.
Note:
Literaturverz. S. [363] - 364
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Title Page; Copyright Page; Preface; Notation; Table of Contents; 1 Introductory Notions; 1.1 Definitions and First Theorems; 1.2 Cosets and Lagrange's Theorem; 1.3 Automorphisms; 2 Normal Subgroups, Conjugation and Isomorphism Theorems; 2.1 Product of Subgroups; 2.2 Normal Subgroups and Quotient Groups; 2.3 Conjugation; 2.4 Normalizers and Centralizers of Subgroups; 2.5 H¨older's Program; 2.6 Direct Products; 2.7 Semidirect Products; 2.8 Symmetric and Alternating Groups; 2.9 The Derived Group; 3 Group Actions and Permutation Groups; 3.1 Group actions; 3.2 The Sylow Theorem
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3.3 Burnside's Formula and Permutation Characters3.4 Induced Actions; 3.5 Permutations Commuting with an Action; 3.6 Automorphisms of Symmetric Groups; 3.7 Permutations and Inversions; 3.8 Some Simple Groups; 3.8.1 The Simple Group of Order 168; 3.8.2 Projective Special Linear Groups; 4 Generators and Relations; 4.1 Generating Sets; 4.2 The Frattini Subgroup; 4.3 Finitely Generated Abelian Groups; 4.4 Free abelian groups; 4.5 Projective and Injective Abelian Groups; 4.6 Characters of Abelian Groups; 4.7 Free Groups; 4.8 Relations; 4.8.1 Relations and simple Groups
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4.9 Subgroups of Free Groups4.10 The Word Problem; 4.11 Residual Properties; 5 Nilpotent Groups and Solvable Groups; 5.1 Central Series and Nilpotent Groups; 5.2 p-Nilpotent Groups; 5.3 Fusion; 5.4 Fixed-Point-Free Automorphisms and Frobenius Groups; 5.5 Solvable Groups; 6 Representations; 6.1 Definitions and examples; 6.1.1 Maschke's Theorem; 6.2 Characters; 6.3 The Character Table; 6.3.1 Burnside's Theorem and Frobenius Theorem; 6.3.2 Topological Groups; 7 Extensions and Cohomology; 7.1 Crossed Homomorphisms; 7.2 The First Cohomology Group; 7.3 The Second Cohomology Group
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7.3.1 H1 and Extensions7.3.2 H2(π,A) for π Finite Cyclic; 7.4 The Schur Multiplier; 7.4.1 Projective Representations; 7.4.2 Covering Groups; 7.4.3 M(π) and Presentations of π; 8 Solution to the exercises; 8.1 Chapter 1; 8.2 Chapter 2; 8.3 Chapter 3; 8.4 Chapter 4; 8.5 Chapter 5; 8.6 Chapter 6; References; Index; Collana Unitext - LaMatematica per il 3+2;
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Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 884702420X
Additional Edition:
Erscheint auch als Druck-Ausgabe Groups : An Introduction to Ideas and Methods of the Theory of Groups
Language:
English
Subjects:
Mathematics
DOI:
10.1007/978-88-470-2421-2
URL:
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