A Course on Finite Groups (Universitext)

A Course on Finite Groups introduces the fundamentals of group theory to advanced undergraduate and beginning graduate students. Based on a series of lecture courses developed by the author over many years, the book starts with the basic definitions and examples and develops the theory to the point where a number of classic theorems can be proved. The topics covered include: group constructions; homomorphisms and isomorphisms; actions; Sylow theory; products and Abelian groups; series; nilpotent and soluble groups; and an introduction to the classification of the finite simple groups. A number of groups are described in detail and the reader is encouraged to work with one of the many computer algebra packages available to construct and experience "actual" groups for themselves in order to develop a deeper understanding of the theory and the significance of the theorems. Numerous problems, of varying levels of difficulty, help to test understanding. A brief resumé of the basic set theory and number theory required for the text is provided in an appendix, and a wealth of extra resources is available online at www.springer.com, including: hints and/or full solutions to all of the exercises; extension material for many of the chapters, covering more challenging topics and results for further study; and two additional chapters providing an introduction to group representation theory. 1848828888 1 Preface 6 Prerequisites 7 Plan of the Book 7 Further Reading 8 Contents 10 Web Contents 12 Chapter 1 14 Introduction-The Group Concept 14 Group Examples 15 Abstract Groups and Representations 17 Classes of Groups 18 Finite Abelian Groups 18 Finite Non-Abelian Groups 18 Infinite Abelian Groups 19 Infinite Non-Abelian Groups 19 Summary of the Book 19 Chapter 2-Elementary Group Properties 19 Chapter 3-Group Construction and Representation 19 Chapter 4-Homomorphisms 20 Chapter 5-Action and the Orbit-stabiliser Theorem 20 Chapter 6-p-Groups and Sylow Theory 20 Chapter 7-Products and Abelian Groups 20 Chapter 8-Groups of Order 24, Three Examples 20 Chapters 9-Series, Jordan-Hölder Theorem and the Extension Problem 21 Chapter 10-Nilpotency 21 Chapter 11-Solubility 21 Chapter 12-Simple Groups of Order Less than 10000 21 Appendix A-Set Theory and Appendix B-Number Theory 21 Appendices C, D and E 21 Web Chapter 13-Representation and Characters 21 Web Chapter 14-Character Tables, and Theorems of Burnside and Frobenius 22 Web Solution Appendix 22 Problems 22 `Actual groups' 22 Computers in Group Theory 22 Chapter 2 24 Elementary Group Properties 24 Basic Definitions 24 Isomorphism-A Preliminary Note 30 Isomorphism Class 30 Examples 30 Number Systems 31 Modular Arithmetic 31 Product Groups 32 Matrix Groups 32 Symmetries of Geometric Objects 32 Permutations 33 Examples from Analysis 34 Free Groups and Presentations 34 Elliptic Curves 35 Examples from Topology 36 Examples from the Physical Sciences 36 Subgroups, Cosets and Lagrange's Theorem 37 Cosets and Lagrange's Theorem 40 Normal Subgroups 43 Subgroup Lattices 45 Problems 47 Chapter 3 54 Group Construction and Representation 54 Permutations 55 Even and Odd Permutations 60 Permutation Groups 61 Alternating Groups 63 Matrix Groups 65 Subgroups of GLn(q) 67 Group Presentation 68 Problems 72 Chapter 4 79 Homomorphisms 79 Maps-Left or Right 80 Homomorphisms and Isomorphisms 81 Factor Groups 84 Isomorphism Theorems 86 Correspondence Theorem 90 Cyclic Groups 92 Automorphism Groups 93 Problems 96 Chapter 5 103 Action and the Orbit-Stabiliser Theorem 103 Actions 104 Restricted Actions 110 Three Important Examples 111 Coset Action 112 Centralisers and Class Equations 113 Normaliser 117 Problems 119 Chapter 6 124 p-Groups and Sylow Theory 124 Finite p-Groups 125 Non-Abelian Groups of Order 8 129 Sylow Theory 130 Applications 137 Groups whose Orders have at most Three Prime Factors 137 Frattini Argument and Nilpotent Groups 139 Problems 142 Chapter 7 149 Products and Abelian Groups 149 Direct Products 150 Uniqueness of Representation 153 Finite Abelian Groups 156 Semi-direct Products 161 Wreath Product 166 Groups of Order 12 166 Problems 169 Chapter 8 174 Groups of Order 24 Three Examples 174 Symmetric Group S4 174 Subgroups of S4 175 Sylow Subgroups 175 Remaining Subgroups 176 Special Linear Group SL2(3) 181 Subgroups of SL2(3) 182 Sylow and Remaining Subgroups 183 Exceptional Group E 186 Subgroups of E 187 Sylow and Non-cyclic Subgroups 187 Problems 192 Chapter 9 195 Series, Jordan-Hölder Theorem and the Extension Problem 195 Composition Series and the Jordan-Hölder Theorem 196 Jordan-Hölder Theorem 198 Extension Problem 204 Factor Pairs 205 Construction of a Group Extension 208 Cyclic Extensions 211 Problems 213 Chapter 10 216 Nilpotency 216 Nilpotent Groups 217 Central Series 217 Nilpotent Groups 219 Frattini and Fitting Subgroups 224 Fitting Subgroup 228 Problems 230 Chapter 11 235 Solubility 235 Soluble Groups 236 Derived Series 240 Minimal Normal Subgroup 241 Hall's Theorems and Solubility Conditions 242 Solubility Conditions 247 Problems 250 Chapter 12 254 Simple Groups of Order Less than 10000 254 Steiner Systems 255 Linear Groups 259 Finite Fields 259 2-Dimensional Linear Groups 260 Groups Ln(q) with n > 2 266 Unitary Groups 270 Mathieu Groups 272 Problems 275 Appendices A to E 281 Set Theory 281 Sets and Maps 281 Relations, Functions and Maps 283 Problems 286 Number Theory 288 Divisibility and the Euclidean Algorithm 288 Congruences 289 Primitive Roots 291 Problems 292 Data on Groups of Order 24 293 Notes on Tables C.1 to C.4 295 Numbers of Groups with Order up to 520 297 Representations of L2(q) with Order < 10000 299 Bibliography 300 Notation Index 304 1-Symbol Index 304 2-Notation for Classes of Groups 306 3-Notation for Individual Groups 307 Index 308 A Course on Finite Groups introduces the fundamentals of group theory to advanced undergraduate and beginning graduate students. Based on a series of lecture courses developed by the author over many years, the book starts with the basic definitions and examples and develops the theory to the point where a number of classic theorems can be proved. The topics covered include: Lagrange's theorem; group constructions; homomorphisms and isomorphisms; actions; Sylow theory, products and Abelian groups; series, and nilpotent and soluble groups; and an introduction to the classification of the finite simple groups. A number of groups are described in detail and the reader is encouraged to work with one of the many computer algebra packages available to construct and experience'actual'groups for themselves in order to develop a deeper understanding of the theory and the significance of the theorems. Numerous exercises, of varying levels of difficulty, help to test understanding. Front Matter....Pages I-XII Introduction—The Group Concept....Pages 1-10 Elementary Group Properties....Pages 11-40 Group Construction and Representation....Pages 41-65 Homomorphisms....Pages 67-90 Action and the Orbit–Stabiliser Theorem....Pages 91-111 p -Groups and Sylow Theory....Pages 113-137 Products and Abelian Groups....Pages 139-163 Groups of Order 24 Three Examples....Pages 165-185 Series, Jordan–Hölder Theorem and the Extension Problem....Pages 187-207 Nilpotency....Pages 209-227 Solubility....Pages 229-247 Simple Groups of Order Less than 10000....Pages 249-275 Appendices A to E....Pages 277-295 Back Matter....Pages 297-311 Introduces the richness of group theory to advanced undergraduate and graduate students, concentrating on the finite aspects. Provides a wealth of exercises and problems to support self-study. Additional online resources on more challenging and more specialised topics can be used as extension material for courses, or for further independent study.