A unique monograph that will become the standard reference
Covers the area from its origins to current research
Large bibliography
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 information that will be especially useful as a reference work for researchers in the field.
John C. Lennox, Research Fellow, Green College and Visiting Fellow, the Mathematical Institute, Oxford University, and Derek J. S. Robinson, Professor of Mathematics, University of Illinois, Urbana, Illinois, USA
Readership: Graduate students and researchers in algebra and group theory
Author(s): John C. Lennox, Derek J. S. Robinson
Series: Oxford Mathematical Monographs
Edition: 1
Publisher: Oxford University Press
Year: 2004
Language: English
Pages: C, xvi, 342
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
