Presentations of Groups (London Mathematical Society Student Texts, Series Number 15)

Emphasizing computational techniques, this book provides an accessible and lucid introduction to combinatorial group theory. Rigorous proofs of all theorems and a light, informal style make Presentations of Groups a self-contained combinatorics class. Numerous and diverse exercises provide readers with a thorough overview of the subject. While catering to combinatorics beginners, this book also includes the frontiers of research, and explains software packages such as GAP, MAGMA, and QUOTPIC. This new edition has been revised throughout, including new exercises and an additional chapter on proving certain groups are infinite. Aimed at advanced undergraduates, this book will be a resource for graduate students and researchers. Contents......Page 4 Preface......Page 8 1.1 Definition and elementary properties......Page 9 1.2 Existence of F(X)......Page 12 1.3 Further properties of F(X)......Page 15 Exercises......Page 19 2.1 The well-ordering of F......Page 22 2.2 The Schreier transversal......Page 24 2.3 The Schreier generators......Page 26 2.4 Decomposition of the set A......Page 27 2.5 Freeness of the generators B......Page 28 2.6 Conclusion......Page 30 Exercises......Page 32 3.1 The finitely-generated case......Page 34 3.2 Example 1......Page 40 3.3 The general case......Page 41 3.4 Further applications......Page 42 Exercise......Page 47 4.1 Basic concepts......Page 49 4.2 Induced homomorphisms......Page 50 4.3 Direct products......Page 52 4.4 Tietze transformations......Page 53 4.5 van Kampen diagrams......Page 59 Exercises......Page 63 5.1 The quaternions......Page 66 5.2 The Heisenberg group......Page 68 5.3 Symmetric groups......Page 69 5.4 Semi-direct products......Page 72 5.5 Groups of symmetries......Page 74 5.6 Polynomials under substitution......Page 76 5.7 The rational numbers......Page 77 Exercises......Page 80 6.1 Groups-made-abelian......Page 82 6.2 Free abelian groups......Page 83 6.3 Change of generators......Page 86 6.4 The invariant factor theorem for matrices......Page 88 6.5 The basis theorem......Page 91 Exercises......Page 93 7. Finite Groups with few Relations......Page 95 7.1 Metacyclic groups......Page 96 7.2 Interesting groups with three generators......Page 100 7.3 Cyclically-presented groups......Page 103 Exercises......Page 105 8.1 The basic method......Page 108 8.2 A refinement......Page 116 Exercises......Page 122 9.1 The method......Page 124 9.2 Alternating groups......Page 126 9.3 Braid groups......Page 127 9.4 von Dyck groups......Page 129 9.5 Triangle groups......Page 135 9.6 Free products......Page 136 9.7 HNN-extensions......Page 137 9.8 The Schur multiplicator......Page 140 Exercises......Page 141 10.1 Basic concepts......Page 144 10.2 The main theorem......Page 146 10.3 Special cases......Page 149 10.4 Finite p-groups......Page 151 Exercises......Page 152 11.1 G-modules......Page 154 11.2 The augmentation ideal......Page 157 11.3 Derivations......Page 159 11.4 Free differential calculus......Page 161 11.5 The fundamental isomorphism......Page 163 Exercises......Page 167 12. An Algorithm for N/N'......Page 168 12.2 The proof......Page 169 12.3 Example......Page 170 Exercises......Page 175 13.1 Review of elementary properties......Page 176 13.2 Power-commutator presentations......Page 178 13.3 mod p modules......Page 181 Exercises......Page 185 14.1 The algorithm......Page 186 14.2 An example......Page 188 14.3 An improvement......Page 192 Exercises......Page 193 15.1 The proof......Page 194 15.2 An example......Page 196 15.3 Related results......Page 197 Exercises......Page 198 16.1 Dimension subgroups......Page 199 16.2 The Gaschütz-Newman formulae......Page 201 16.3 Newman's criterion......Page 204 16.4 Fibonacci update......Page 207 Exercises......Page 208 Guide to the literature and references......Page 209 Index......Page 218 Dramatis personae......Page 222 The aim of this book is to provide an introduction to combinatorial group theory. Any reader who has completed first courses in linear algebra, group theory and ring theory will find this book accessible. The emphasis is on computational techniques but rigorous proofs of all theorems are supplied. This new edition has been revised throughout, including new exercises and an additional chapter on proving that certain groups are infinite.