The Theory of Infinite Soluble Groups

by Lennox, John C.; Robinson, Derek J. S.

• ISBN13: 9780198507284
• ISBN10: 0198507283
• Format: Hardcover
• Copyright: 2004-10-21
• Publisher: Clarendon Press

Summary

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.

Table of Contents

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