This seminal, much-cited account begins with a fairly elementary exposition of basic concepts and a discussion of factor groups and subgroups. The topics of Nielsen transformations, free and amalgamated products, and commutator calculus receive detailed treatment. The concluding chapter surveys word, conjugacy, and related problems; adjunction and embedding problems; and more. Second, revised 1976 edition.

Technical Remarks

Chapter 1 Basic Concepts

1.1 Introduction

1

(11)

1.2 Construction of groups from generators and defining relators

12

(12)

1.3 Dehn's fundamental problems

24

(9)

1.4 Definition and elementary properties of free groups

33

(15)

1.5 Tietze transformations

48

(8)

1.6 Graph of a group

56

(15)

Chapter 2 Factor Groups and Subgroups

2.1 Factor groups

71

(3)

2.2 Verbal subgroups and reduced free groups

74

(12)

2.3 Presentations of subgroups (The Reidemeister-Schreier method)

86

(18)

2.4 Subgroups of free groups

104

(16)

Chapter 3 Nielsen Transformations

3.1 Introduction

120

(1)

3.2 A reduction process

121

(19)

3.3 The commutator quotient group

140

(13)

3.4 A test for isomorphism

153

(9)

3.5 The automorphism group Φn of free groups

162

(7)

3.6 Free automorphisms and free isomorphisms

169

(3)

3.7 Braid groups and mapping class groups

172

(8)

Chapter 4 Free Products and Free Products with Amalgamations

4.1 Free products

180

(17)

4.2 Free product with amalgamated subgroups

197

(30)

4.3 Subgroup theorems for free and amalgamated products

227

(25)

4.4 Groups with one defining relator

252

(35)

Chapter 5 Commutator Calculus

5.1 Introduction

287

(1)

5.2 Commutator identities

288

(4)

5.3 The lower central series

292

(6)

5.4 Some freely generated graded algebras

298

(10)

5.5 A mapping of a free group into Α(Ζ, r)

308

(9)

5.6 Lie elements and basis theorems

317

(19)

5.7 The lower central series of free groups

336

(13)

5.8 Some applications

349

(8)

5.9 Identities

357

(11)

5.10 The Baker-Hausdorff formula

368

(5)

5.11 Power relations and commutator relations

373

(6)

5.12 Burnside's problem, Exponents 3 and 4

379

(7)

5.13 Burnside's problem, Report on e > 4

386

(2)

5.14 Topological aspects

388

(5)

5.15 Free differential calculus

393

(3)

Chapter 6 Introduction to Some Recent Developments

6.1 Word, conjugacy, and related problems

396

(5)

6.2 Adjunction and embedding problems

401

(5)

6.3 Varieties of groups

406

(4)

6.4 Products of groups

410

(3)

6.5 Residual and Hopfian properties

413

(8)

References

421

(15)

List of Theorems, Corollaries, and Definitions

436

(2)

List of Symbols and Abbreviations

438

(3)

Index

441

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Combinatorial Group Theory Presentations of Groups in Terms of Generators and Relations

9780486438306

0486438309

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