The Theory of Infinite Soluble Groups

The central concept in this monograph is that of a soluble group - a group which is built up from abelian groups by repeatedly forming group extensions. It covers all the major areas, including finitely generated soluble groups, soluble groups of finite rank, modules over group rings,algorithmic problems, applications of cohomology, and finitely presented groups, whilst remaining fairly strictly within the boundaries of soluble group theory. An up-to-date survey of the area aimed at research students and academic algebraists and group theorists, it is a compendium of informationthat will be especially useful as a reference work for researchers in the field.

List of Symbols

xiii

Introduction

xv

1 Basic results on soluble and nilpotent groups

1

(28)

1.1 Definition and elementary properties of soluble groups

1

(3)

Commutators and Commutator subgroups

1.2 Definition and elementary properties of nilpotent groups

4

(10)

Stability groups, Fitting's Theorem and the Fitting subgroup

Inheritance properties of the lower central series

1.3 Polycyclic groups

14

(9)

Frattini subgroups

Examples of polycyclic groups

1.4 Soluble groups with the minimal condition

23

(3)

Constructing Cernikov groups

Nilpotent groups with min

1.5 Soluble groups with the minimal condition on normal subgroups

26

(3)

2 Nilpotent groups

29

(18)

2.1 Extraction of roots in nilpotent groups

29

(5)

Embedding in radicable groups-the proof of Mal'cev's Theorem

Uniqueness of the completion

Extension to locally nilpotent groups

2.2 Basic commutators

34

(4)

The collection process

2.3 The theory of isolators

38

(9)

3 Soluble linear groups

47

(13)

3.1 Mal'cev's Theorem

47

(7)

Triangularizable, unitriangularizable, and diagonalizable groups

3.2 Soluble Z-linear groups

54

(3)

Torsion groups of automorphisms of Cernikov groups

3.3 The linearity of polycyclic groups

57

(3)

4 The theory of finitely generated soluble groups I

60

(23)

4.1 Embedding in finitely generated soluble groups

60

(2)

4.2 The maximal condition on normal subgroups

62

(3)

4.3 Residual finiteness. The Artin-Rees Lemma

65

(8)

4.4 The Fitting and Frattini subgroups in finitely generated soluble groups

73

(4)

4.5 Counterexamples

77

(4)

4.6 Engel elements in soluble groups

81

(2)

5 Soluble groups of finite rank

83

(22)

5.1 The ranks of an abelian group

83

(7)

Classes of soluble groups of finite rank

Weak maximal and minimal conditions

5.2 Structure theorems for soluble groups of finite rank

90

(6)

Polyrational groups

The Prüfer rank of a polyrational group

Chief factors and maximal subgroups

5.3 Residual finiteness of soluble groups of finite rank

96

(9)

Frattini subgroups of soluble minimax groups

6 Finiteness conditions on abelian subgroups

105

(16)

6.1 Chain conditions on abelian subgroups

105

(4)

6.2 Finite rank conditions on abelian subgroups

109

(4)

6.3 Chain conditions on subnormal or ascendant abelian subgroups

113

(8)

7 The theory of finitely generated soluble groups II

121

(22)

7.1 Simple modules over polycyclic groups

121

(9)

7.2 Artin-Rees properties and residual finiteness

130

(2)

7.3 Frattini properties of finitely generated abelian-by-polycyclic groups

132

(6)

7.4 Just non-polycyclic groups

138

(5)

8 Centrality in finitely generated soluble groups

143

(15)

8.1 The centrality theorems

143

(3)

8.2 The Fan Out Lemma

146

(3)

8.3 Proofs of the main centrality theorems

149

(6)

8.4 Bryant's verbal topology

155

(1)

8.5 Centrality in finitely generated abelian-by-polycyclic groups

156

(2)

9 Algorithmic theories of finitely generated soluble groups

158

(33)

9.1 The classical decision problems of group theory

158

(4)

Negative results for finitely presented soluble groups

9.2 Algorithms for polycyclic; groups

162

(13)

Polycyclic presentations

Virtually polycyclic subgroups of GLn (Z)

9.3 Algorithms for finitely generated soluble minimax groups

175

(3)

9.4 Submodule computability

178

(7)

9.5 Algorithms for finitely generated metabelian groups

185

(6)

The conjugacy problem

Further algorithms

10 Cohomological methods in infinite soluble group theory

191

(49)

10.1 The cohomology groups in group theory

191

(12)

The Gruenberg Resolution

The homology of polycyclic groups

Group theoretic interpretations

10.2 Soluble groups with finite (co)homological dimensions

203

(7)

10.3 Cohomological vanishing theorems for nilpotent groups

210

(10)

Bounded (co)homology groups

10.4 Applications to infinite soluble groups

220

(9)

Nilpotent supplements

Nearly maximal subgroups

Complete abelian-by-nilpotent groups

Soluble products of polycyclic groups

10.5 Kropholler's theorem on soluble minimax groups

229

(11)

11 Finitely presented soluble groups

240

(35)

11.1 Some finitely presented and infinitely presented soluble groups

240

(5)

11.2 Constructible soluble groups

245

(6)

Structure of constructible soluble groups

Subgroups of constructible soluble groups

11.3 Embedding in finitely presented metabelian groups

251

(5)

11.4 Structural properties of finitely presented soluble groups

256

(6)

Application to coherent soluble groups

Finitely presented nilpotent-by-cyclic groups

11.5 The Bieri-Strebel invariant

262

(13)

Valuation spheres

Tame modules

Finitely presented centre-by-metabelian groups

Characterizing finitely presented metabelian groups

12 Subnormality and solubility

275

(15)

12.1 Soluble groups and the subnormal intersection property

275

(4)

Soluble groups with bounded subnormal defects

12.2 Groups with every subgroup subnormal

279

(3)

Reductions

The derived series

12.3 Torsion-free groups with all subgroups subnormal-solubility

282

(3)

12.4 Torsion-free groups with all subgroups subnormal-nilpotence

285

(2)

12.5 Torsion groups with all subgroups subnormal recent developments

287

(3)

Bibliography

290

(45)

Index of Authors

335

(3)

Index

338

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The Theory of Infinite Soluble Groups

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