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

by Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald

• ISBN13: 9780486438306
• ISBN10: 0486438309
• Edition: 2nd
• Format: Paperback
• Copyright: 2004-11-12
• Publisher: Dover Publications

Summary

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.

Table of Contents

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