The theory of infinite soluble groups

359 Pages·2004·1.581 MB·English

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

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JohnC.LennoxandDerekJ.S.Robinson:Thetheoryofinfinitesolublegroups
The Theory of Infinite
Soluble Groups
JOHN C. LENNOX
Green College, University of Oxford, England
and
DEREK J. S. ROBINSON
University of Illinois, Urbana, Illinois, USA
CLARENDON PRESS • OXFORD
2004
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In memory of Philip Hall
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“To leave no rubs nor botches in the work ...”
Shakespeare, Macbeth,
Act 3, Scene 1.
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CONTENTS
List of Symbols xiii
Introduction xv
1 Basic results on soluble and nilpotent groups 1
1.1 Definition and elementary properties of soluble groups 1
Commutators and Commutator subgroups
1.2 Definition and elementary properties of nilpotent groups 4
Stability groups, Fitting’s Theorem and the Fitting
subgroup
Inheritance properties of the lower central series
1.3 Polycyclic groups 14
Frattini subgroups
Examples of polycyclic groups
1.4 Soluble groups with the minimal condition 23
Constructing Cˇernikov groups
Nilpotent groups with min
1.5 Soluble groups with the minimal condition on normal subgroups 26
2 Nilpotent groups 29
2.1 Extraction of roots in nilpotent groups 29
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
The collection process
2.3 The theory of isolators 38
3 Soluble linear groups 47
3.1 Mal’cev’s Theorem 47
Triangularizable, unitriangularizable, and
diagonalizable groups
3.2 Soluble Z-linear groups 54
Torsion groups of automorphisms of Cˇernikov groups
3.3 The linearity of polycyclic groups 57
4 The theory of finitely generated soluble groups I 60
4.1 Embedding in finitely generated soluble groups 60
4.2 The maximal condition on normal subgroups 62
ix