A Course in Group Theory / Edition 1

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Overview


The classification of the finite simple groups is one of the major intellectual achievements of this century, but it remains almost completely unknown outside of the mathematics community. This introduction to group theory is also an attempt to make this important work better known. Emphasizing classification themes throughout, the book gives a clear and comprehensive introduction to groups and covers all topics likely to be encountered in an undergraduate course. Introductory chapters explain the concepts of group, subgroup and normal subgroup, and quotient group. The homomorphism and isomorphism theorems are explained, along with an introduction to G-sets. Subsequent chapters deal with finite abelian groups, the Jordan-Holder theorem, soluble groups, p-groups, and group extensions. The numerous worked examples and exercises in this excellent and self-contained introduction will also encourage undergraduates (and first year graduates) to further study.
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Product Details

  • ISBN-13: 9780198534594
  • Publisher: Oxford University Press, USA
  • Publication date: 4/28/1996
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 296
  • Product dimensions: 9.32 (w) x 6.19 (h) x 0.61 (d)

Table of Contents

1 Definitions and examples 1
2 Maps and relations on sets 8
3 Elementary consequences of the definitions 18
4 Subgroups 30
5 Cosets and Lagrange's Theorem 38
6 Error-correcting codes 49
7 Normal subgroups and quotient groups 59
8 The Homomorphism Theorem 68
9 Permutations 77
10 The Orbit-Stabiliser Theorem 89
11 The Sylow Theorems 98
12 Applications of Sylow theory 106
13 Direct products 112
14 The classification of finite abelian groups 120
15 The Jordan-Holder Theorem 128
16 Composition factors and chief factors 137
17 Soluble groups 146
18 Examples of soluble groups 155
19 Semidirect products and wreath products 163
20 Extensions 174
21 Central and cyclic extensions 183
22 Groups with at most 31 elements 192
23 The projective special linear groups 202
24 The Mathieu groups 213
25 The classification of finite simple groups 222
A Prerequisites from number theory and linear algebra 234
B Groups of order < 32 238
C Solutions to exercises 243
Bibliography 275
Index 277
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