Presentations of Groups

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.

Preface

xi

1. Free Groups

1

(13)

1.1 Definition and elementary properties

1

(3)

1.2 Existence of F(X)

4

(3)

1.3 Further properties of F(X)

7

(4)

Exercises

11

(3)

2. Schreier's Method

14

(12)

2.1 The well-ordering of F

14

(2)

2.2 The Schreier transversal

16

(2)

2.3 The Schreier generators

18

(1)

2.4 Decomposition of the set A

19

(1)

2.5 Freeness of the generators B

20

(2)

2.6 Conclusion

22

(2)

Exercises

24

(2)

3. Nielsen's Method

26

(15)

3.1 The finitely-generated case

26

(6)

3.2 Example 1

32

(1)

3.3 The general case

33

(1)

3.4 Further applications

34

(5)

Exercise

39

(2)

4. Free Presentations of Groups

41

(17)

4.1 Basic concepts

41

(1)

4.2 Induced homomorphisms

42

(2)

4.3 Direct products

44

(1)

4.4 Tietze transformations

45

(6)

4.5 van Kampen diagrams

51

(4)

Exercises

55

(3)

5. Some Popular Groups

58

(16)

5.1 The quaternions

58

(2)

5.2 The Heisenberg group

60

(1)

5.3 Symmetric groups

61

(3)

5.4 Semi-direct products

64

(2)

5.5 Groups of symmetries

66

(2)

5.6 Polynomials under substitution

68

(1)

5.7 The rational numbers

69

(3)

Exercises

72

(2)

6. Finitely-generated Abelian Groups

74

(13)

6.1 Groups-made-abelian

74

(1)

6.2 Free abelian groups

75

(3)

6.3 Change of generators

78

(2)

6.4 The invariant factor theorem for matrices

80

(3)

6.5 The basis theorem

83

(2)

Exercises

85

(2)

7. Finite Groups with few Relations

87

(13)

7.1 Metacyclic groups

88

(4)

7.2 Interesting groups with three generators

92

(3)

7.3 Cyclically-presented groups

95

(2)

Exercises

97

(3)

8. Coset Enumeration

100

(16)

8.1 The basic method

100

(8)

8.2 A refinement

108

(6)

Exercises

114

(2)

9. Presentations of Subgroups

116

(20)

9.1 The method

116

(2)

9.2 Alternating groups

118

(1)

9.3 Braid groups

119

(2)

9.4 von Dyck groups

121

(6)

9.5 Triangle groups

127

(1)

9.6 Free products

128

(1)

9.7 HNN-extensions

129

(3)

9.8 The Schur multiplicator

132

(1)

Exercises

133

(3)

10. Presentations of Group Extensions

136

(10)

10.1 Basic concepts

136

(2)

10.2 The main theorem

138

(3)

10.3 Special cases

141

(2)

10.4 Finite p-groups

143

(1)

Exercises

144

(2)

11. Relation Modules

146

(14)

11.1 G-modules

146

(3)

11.2 The augmentation ideal

149

(2)

11.3 Derivations

151

(2)

11.4 Free differential calculus

153

(2)

11.5 The fundamental isomorphsim

155

(4)

Exercises

159

(1)

12. An Algorithm for N/N'

160

(8)

12.1 The Jacobian

161

(1)

12.2 The proof

161

(1)

12.3 Example

162

(5)

Exercises

167

(1)

13. Finite p-groups

168

(10)

13.1 Review of elementary properties

168

(2)

13.2 Power-commutator presentations

170

(3)

13.3 mod p modules

173

(4)

Exercises

177

(1)

14. The Nilpotent Quotient Algorithm

178

(8)

14.1 The algorithm

178

(2)

14.2 An example

180

(4)

14.3 An improvement

184

(1)

Exercises

185

(1)

15. The Golod-Shafarevich Theorem

186

(5)

15.1 The proof

186

(2)

15.2 An example

188

(1)

15.3 Related results

189

(1)

Exercises

190

(1)

16. Proving some Groups Infinite

191

(10)

16.1 Dimension subgroups

191

(2)

16.2 The Gaschutz-Newman formulae

193

(3)

16.3 Newman's criterion

196

(3)

16.4 Fibonacci update

199

(1)

Exercises

200

(1)

Guide to the literature and references

201

(10)

Index

211

(5)

Dramatis personae

216

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Presentations of Groups

Summary

Table of Contents

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