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Intro -- Preface -- Acknowledgments -- Contents -- Notations -- 1 Commutator Calculus -- 1.1 The Center of a Group -- 1.1.1 Conjugates and Central Elements -- 1.1.2 Examples Involving the Center -- 1.1.3 Central Subgroups and the Centralizer -- 1.1.4 The Center of a p-Group -- 1.2 The Commutator of Group Elements -- 1.3 Commutator Subgroups -- 1.3.1 Properties of Commutator Subgroups -- 1.3.2 The Normal Closure -- References -- 2 Introduction to Nilpotent Groups -- 2.1 The Lower and Upper Central Series -- 2.1.1 Series of Subgroups -- 2.1.2 Definition of a Nilpotent Group -- 2.1.3 The Lower Central Series -- 2.1.4 The Upper Central Series -- 2.1.5 Comparing Central Series -- 2.2 Examples of Nilpotent Groups -- 2.2.1 Finite p-Groups -- 2.2.2 An Example Involving Rings -- 2.3 Elementary Properties of Nilpotent Groups -- 2.3.1 Establishing Nilpotency by Induction -- 2.3.2 A Theorem on Root Extraction -- 2.3.3 The Direct Product of Nilpotent Groups -- 2.3.4 Subnormal Subgroups -- 2.3.5 The Normalizer Condition -- 2.3.6 Products of Normal Nilpotent Subgroups -- 2.4 Finite Nilpotent Groups -- 2.5 The Tensor Product of the Abelianization -- 2.5.1 The Three Subgroup Lemma -- 2.5.2 The Epimorphism ZnAb(G) →γnG/γn + 1G -- 2.5.3 Property P -- 2.5.4 The Hirsch-Plotkin Radical -- 2.5.5 An Extension Theorem for Nilpotent Groups -- 2.6 Finitely Generated Torsion Nilpotent Groups -- 2.6.1 The Torsion Subgroup of a Nilpotent Group -- 2.7 The Upper Central Subgroups and Their Factors -- 2.7.1 Intersection of the Center and a Normal Subgroup -- 2.7.2 Separating Points in a Group -- References -- 3 The Collection Process and Basic Commutators -- 3.1 The Collection Process -- 3.1.1 Weighted Commutators -- 3.1.2 The Collection Process for Weighted Commutators -- 3.1.3 Basic Commutators -- 3.1.4 The Collection Process For Arbitrary Group Elements -- 3.2 The Collection Formula -- 3.2.1 Preliminary Examples -- 3.2.2 The Collection Formula and Applications -- 3.3 A Basis Theorem -- 3.3.1 Groupoids and Basic Sequences -- 3.3.2 Basic Commutators Revisited -- 3.3.3 Lie Rings and Basic Lie Products -- 3.3.4 The Commutation Lie Ring -- 3.3.5 The Magnus Embedding -- 3.4 Proof of the Collection Formula -- References -- 4 Normal Forms and Embeddings -- 4.1 The Hall-Petresco Words -- 4.1.1 m-Fold Commutators -- 4.1.2 A Collection Process -- 4.1.3 The Hall-Petresco Words -- 4.2 Normal Forms and Mal''cev Bases -- 4.2.1 The Structure of a Finitely Generated Nilpotent Group -- 4.2.2 Mal''cev Bases -- 4.3 The R-Completion of a Finitely Generated Torsion-Free Nilpotent Group -- 4.3.1 R-Completions -- 4.3.2 Mal''cev Completions -- 4.4 Nilpotent R-Powered Groups -- 4.4.1 Definition of a Nilpotent R-Powered Group -- 4.4.2 Examples of Nilpotent R-Powered Groups -- 4.4.3 R-Subgroups and Factor R-Groups -- 4.4.4 R-Morphisms -- 4.4.5 Direct Products -- 4.4.6 Abelian R-Groups -- 4.4.7 Upper and Lower Central Series -- 4.4.8 Tensor Product of the Abelianization -- 4.4.9 Condition Max-R -- 4.4.10 An R-Series of a Finitely R-Generated Nilpotent R-Powered Group -- 4.4.11 Free Nilpotent R-Powered Groups -- 4.4.12 ω-Torsion and R-Torsion -- 4.4.13 Finite ω-Type and Finite Type -- 4.4.14 Order, Exponent, and Power-Commutativity -- References -- 5 Isolators, Extraction of Roots, and P-Localization -- 5.1 The Theory of Isolators -- 5.1.1 Basic Properties of P-Isolated Subgroups -- 5.1.2 The P-Isolator -- 5.1.3 P-Equivalency -- 5.1.4 P-Isolators and Transfinite Ordinals -- 5.2 Extraction of Roots -- 5.2.1 UP-Groups -- 5.2.2 UP-Groups and Quotients -- 5.2.3 P-Torsion-Free Locally Nilpotent Groups -- 5.2.4 Upper Central Subgroups of UP-Groups -- 5.2.5 Extensions of UP-Groups -- 5.2.6 P-Radicable and Semi-P-Radicable Groups -- 5.2.7 Extensions of P-Radicable Groups -- 5.2.8 Divisible Groups -- 5.2.9 Nilpotent P-Radicable Groups -- 5.2.10 The Structure of a Radicable Nilpotent Group -- 5.2.11 The Maximal P-Radicable Subgroup -- 5.2.12 Residual Properties -- 5.2.13 DP-Groups -- 5.2.14 Extensions of DP-Groups -- 5.2.15 Some Embedding Theorems -- 5.3 The P-Localization of Nilpotent Groups -- 5.3.1 P-Local Groups -- 5.3.2 Fundamental Theorem of P-Localization of Nilpotent Groups -- 5.3.3 P-Morphisms -- References -- 6 ``The Group Ring of a Class of Infinite Nilpotent Groups'''' by S. A. Jennings -- 6.1 The Group Ring of a Torsion-Free Nilpotent Group -- 6.1.1 Group Rings and the Augmentation Ideal -- 6.1.2 Residually Nilpotent Augmentation Ideals -- 6.1.3 E. Formanek''s Generalization -- 6.2 The Dimension Subgroups -- 6.2.1 Definition and Properties of Dimension Subgroups -- 6.2.2 Faithful Representations -- 6.3 The Lie Algebra of a Finitely Generated Torsion-Free Nilpotent Group -- 6.3.1 Lie Algebras -- 6.3.2 The A-Adic Topology on FG -- 6.3.3 The Group 1 + A -- 6.3.4 The Maps Exp and Log -- 6.3.5 The Lie Algebra Λ -- 6.3.6 The Baker-Campbell-Hausdorff Formula -- 6.3.7 The Lie Algebra of a Nilpotent Group -- 6.3.8 The Group (to. Λ, )to. -- 6.3.9 A Theorem on Automorphism Groups -- 6.3.10 The Mal''cev Correspondence -- References -- 7 Additional Topics -- 7.1 Decision Problems -- 7.1.1 The Word Problem -- 7.1.2 The Conjugacy Problem -- 7.2 The Hopfian Property -- 7.3 The (Upper) Unitriangular Groups -- 7.4 Nilpotent Groups of Automorphisms -- 7.4.1 The Stability Group -- 7.4.2 The IA-Group of a Nilpotent Group -- 7.5 The Frattini and Fitting Subgroups -- 7.5.1 The Frattini Subgroup -- 7.5.2 The Fitting Subgroup -- References -- Index -- .