Combinatorial group theory: Presentations of groups in terms of generators and relations

Wilhelm Magnus, Abraham Karrass, Donald Solitar9780486438306, 0486438309

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.

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Table of contents :
Contents……Page 8
Technical Remarks……Page 11
1.1 Introduction……Page 12
1.2 Construction of groups from generators and defining relators……Page 23
1.3 Dehn’s fundamental problems……Page 35
1.4 Definition and elementary properties of free groups……Page 44
1.5 Tietze transformations……Page 59
1.6 Graph of a group……Page 67
2.1 Factor groups……Page 82
2.2 Verbal subgroups and reduced free groups……Page 85
2.3 Presentations of subgroups (The Reidemeister-Schreier method)……Page 97
2.4 Subgroups of free groups……Page 115
3.1 Introduction……Page 131
3.2 A reduction process……Page 132
3.3 The commutator quotient group……Page 151
3.4 A test for isomorphism……Page 164
3.5 The automorphism group Phi_n of free groups……Page 173
3.6 Free automorphisms and free isomorphisms……Page 180
3.7 Braid groups and mapping class groups……Page 183
4.1 Free products……Page 191
4.2 Free product with amalgamated subgroups……Page 208
4.3 Subgroup theorems for free and amalgamated products……Page 238
4.4 Groups with one defining relator……Page 263
5.1 Introduction……Page 298
5.2 Commutator identities……Page 299
5.3 The lower central series……Page 303
5.4 Some freely generated graded algebras……Page 309
5.5 A mapping of a free group into A(Z, r)……Page 319
5.6 Lie elements and basis theorems……Page 328
5.7 The lower central series of free groups……Page 347
5.8 Some applications……Page 360
5.9 Identities……Page 368
5.10 The Baker-Hausdorff formula……Page 379
5.11 Power relations and commutator relations……Page 384
5.12 Burnside’s problem, Exponents 3 and 4……Page 390
5.13 Burnside’s problem, Report on e > 4……Page 397
5.14 Topological aspects……Page 399
5.15 Free differential calculus……Page 404
6.1 Word, conjugacy, and related problems……Page 407
6.2 Adjunction and embedding problems……Page 412
6.3 Varieties of groups……Page 417
6.4 Products of groups……Page 421
6.5 Residual and Hopfian properties……Page 424
References……Page 431
List of Theorems, Corollaries, and Definitions……Page 446
List of Symbols and Abbreviations……Page 448
Index……Page 450