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.
Author(s): D. L. Johnson
Series: London Mathematical Society student texts 15
Edition: 2nd ed
Publisher: Cambridge University Press
Year: 1997
Language: English
Pages: 223
City: Cambridge, U.K.; New York, NY, USA
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
