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Groups An Introduction to Ideas and Methods of the Theory of Groups

:	Antonio Machì
:	Groups An Introduction to Ideas and Methods of the Theory of Groups
:	Springer Verlag Italia
:	9788847024212
:	1
:	CHF 37.80
:

:	Arithmetik, Algebra
:	English

:	385
:	DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	Preface	5
	Notation	8
	Table of Contents	9
	1 Introductory Notions	12
	1.1 Definitions and First Theorems	12
	1.2 Cosets and Lagrange’s Theorem	35
	1.3 Automorphisms	45
	2 Normal Subgroups, Conjugation and Isomorphism Theorems	49
	2.1 Product of Subgroups	49
	2.2 Normal Subgroups and Quotient Groups	50
	2.3 Conjugation	60
	2.4 Normalizers and Centralizers of Subgroups	69
	2.5 H¨older’s Program	73
	2.6 Direct Products	77
	2.7 Semidirect Products	83
	2.8 Symmetric and Alternating Groups	88
	2.9 The Derived Group	92
	3 Group Actions and Permutation Groups	97
	3.1 Group actions	97
	3.2 The Sylow Theorem	109
	3.3 Burnside’s Formula and Permutation Characters	127
	3.4 Induced Actions	135
	3.5 Permutations Commuting with an Action	138
	3.6 Automorphisms of Symmetric Groups	144
	3.7 Permutations and Inversions	146
	3.8 Some Simple Groups	153
	3.8.1 The Simple Group of Order 168	153
	3.8.2 Projective Special Linear Groups	157
	4 Generators and Relations	164
	4.1 Generating Sets	164
	4.2 The Frattini Subgroup	169
	4.3 Finitely Generated Abelian Groups	173
	4.4 Free abelian groups	179
	4.5 Projective and Injective Abelian Groups	187
	4.6 Characters of Abelian Groups	190
	4.7 Free Groups	192
	4.8 Relations	197
	4.8.1 Relations and simple Groups	201
	4.9 Subgroups of Free Groups	203
	4.10 The Word Problem	207
	4.11 Residual Properties	209
	5 Nilpotent Groups and Solvable Groups	214
	5.1 Central Series and Nilpotent Groups	214
	5.2 p-Nilpotent Groups	232
	5.3 Fusion	238
	5.4 Fixed-Point-Free Automorphisms and Frobenius Groups	241
	5.5 Solvable Groups	246
	6 Representations	262
	6.1 Definitions and examples	262
	6.1.1 Maschke’s Theorem	266
	6.2 Characters	268
	6.3 The Character Table	284
	6.3.1 Burnside’s Theorem and Frobenius Theorem	289
	6.3.2 Topological Groups	294
	7 Extensions and Cohomology	298
	7.1 Crossed Homomorphisms	298
	7.2 The First Cohomology Group	301
	7.3 The Second Cohomology Group	310
	7.3.1 H1 and Extensions	316
	7.3.2 H2(p,A) for p Finite Cyclic	317
	7.4 The Schur Multiplier	321
	7.4.1 Projective Representations	322
	7.4.2 Covering Groups	324
	7.4.3 M(p) and Presentations of p	329
	8 Solution to the exercises	335
	8.1 Chapter 1	335
	8.2 Chapter 2	337
	8.3 Chapter 3	343
	8.4 Chapter 4	354
	8.5 Chapter 5	359
	8.6 Chapter 6	366
	References	371
	Index	373