The theory of infinite soluble groups

Cover
Series Editors
The Theory of Infinite Soluble Groups
Copyright (c) 2004 by John C. Lennox and Derek J. S. Robinson
     ISBN 0 19 850728 3 (Hbk)
dedicated In memory of Philip Hall 1904–1982
CONTENTS
LIST OF SYMBOLS

INTRODUCTION

1  BASIC RESULTS ON SOLUBLE AND NILPOTENT GROUPS
     1.1 Definition and elementary properties of soluble groups
     1.2 Definition and elementary properties of nilpotent groups
     1.3 Polycyclic groups
     1.4 Soluble groups with the minimal condition
     1.5 Soluble groups with the minimal condition on normal subgroups

2  NILPOTENT GROUPS
     2.1 Extraction of roots in nilpotent groups
     2.2 Basic commutators
     2.3 The theory of isolators

3  SOLUBLE LINEAR GROUPS
     3.1 Mal’cev’s Theorem
     3.2 Soluble Z-linear groups
     3.3 The linearity of polycyclic groups

4  THE THEORY OF FINITELY GENERATED SOLUBLE GROUPS I
     4.1 Embedding in finitely generated soluble groups
     4.2 The maximal condition on normal subgroups
     4.3 Residual finiteness
     4.4 The Fitting and Frattini subgroups in finitely generated soluble groups
     4.5 Counterexamples
     4.6 Engel elements in soluble groups

5  SOLUBLE GROUPS OF FINITE RANK
     5.1 The ranks of an abelian group
     5.2 Structure theorems for soluble groups of finite rank
     5.3 Residual finiteness of soluble groups of finite rank

6  FINITENESS CONDITIONS ON ABELIAN SUBGROUPS
     6.1 Chain conditions on abelian subgroups
     6.2 Finite rank conditions on abelian subgroups
     6.3 Chain conditions on subnormal or ascendant abelian subgroups

7  THE THEORY OF FINITELY GENERATED SOLUBLE GROUPS II
     7.1 Simple modules over polycyclic groups
     7.2 Artin–Rees properties and residual finiteness
     7.3 Frattini properties of finitely generated abelian-by-polycyclic groups
     7.4 Just non-polycyclic groups

8  CENTRALITY IN FINITELY GENERATED SOLUBLE GROUPS
     8.1 The centrality theorems
     8.2 The Fan Out Lemma
     8.3 Proofs of the main centrality theorems
     8.4 Bryant’s verbal topology
     8.5 Centrality in finitely generated abelian-by-polycyclic groups

9  ALGORITHMIC THEORIES OF FINITELY GENERATED SOLUBLE GROUPS
     9.1 The classical decision problems of group theory
     9.2 Algorithms for polycyclic groups
     9.3 Algorithms for finitely generated soluble minimax groups
     9.4 Submodule computability
     9.5 Algorithms for finitely generated metabelian groups

10  COHOMOLOGICAL METHODS IN INFINITE SOLUBLE GROUP THEORY
     10.1 The cohomology groups in group theory
     10.2 Soluble groups with finite (co)homological dimensions
     10.3 Cohomological vanishing theorems for nilpotent groups
     10.4 Applications to infinite soluble groups
     10.5 Kropholler’s theorem on soluble minimax groups

11  FINITELY PRESENTED SOLUBLE GROUPS
     11.1 Some finitely presented and infinitely presented soluble groups
     11.2 Constructible soluble groups
     11.3 Embedding in finitely presented metabelian groups
     11.4 Structural properties of finitely presented soluble groups
     11.5 The Bieri–Strebel invariant

12  SUBNORMALITY AND SOLUBILITY
     12.1 Soluble groups and the subnormal intersection property
     12.2 Groups with every subgroup subnormal
     12.3 Torsion-free groups with all subgroups subnormal—solubility
     12.4 Torsion-free groups with all subgroups subnormal—nilpotence
     12.5 Torsion groups with all subgroups subnormal—recent developments

BIBLIOGRAPHY
INDEX OF AUTHORS
INDEX