Lecture Notes in Mathematics

By: Roman Mikhailov, Inder Bir Singh Passi

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Paperback | 16 October 2008

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A fundamental object of study in group theory is the lower central series of groups. Understanding its relationship with the dimension series, which consists of the subgroups determined by the augmentation powers, is a challenging task. This monograph presents an exposition of different methods for investigating this relationship. In addition to group theorists, the results are also of interest to topologists and number theorists. The approach is mainly combinatorial and homological. A novel feature is an exposition of simplicial methods for the study of problems in group theory.

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"The authors of this book present an exposition of different methods related to the theory of the lower central series of groups, the dimension subgroups, the augmentation powers and the derived series. ... This book is very useful for future investigations of the lower central series of groups and the dimension subgroups." (Todor Mollov, Zentralblatt MATH, Vol. 1157, 2009)

"The focus is primarily on homological, homotopical and combinatorial methods for the study of group rings. ... On the whole, the monograph is a welcome addition for those interested in the various aspects of the study of group rings and their applications. A strong point of the present book ... is the numerous examples given in each chapter either to confirm a certain result or as a counterexample of a result under weaker hypothesis." (L. R. Vermani, Mathematical Reviews, Issue 2009 m)

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Lower Central Series	p. 1
Commutator Calculus	p. 1
Residually Nilpotent Groups	p. 2
Generalized Relation Modules	p. 17
k-central Extensions	p. 37
Nilpotent Completion	p. 66
Bousfield-Kan Completion	p. 71
Homological Localization	p. 75
Crossed Modules and Cat[superscript 1]-Groups	p. 84
Dimension Subgroups	p. 101
Groups Without Dimension Property	p. 101
Sjogren's Theorem	p. 119
Fourth Dimension Subgroup	p. 121
Fifth Dimension Subgroup	p. 123
Quasi-varieties of Groups	p. 134
The Quasi-variety D[subscript 4]	p. 137
Dimension Quotients	p. 143
Plotkin's Problems	p. 148
Modular Dimension Subgroups	p. 151
Lie Dimension Subgroups	p. 152
Lie Nilpotency Indices	p. 160
Subgroups Dual to Dimension Subgroups	p. 163
Derived Series	p. 165
Commutator Subgroups of Free Nilpotent Groups	p. 165
E and D-groups	p. 171
Transfinite Derived Series	p. 176
Applications to Asphericity	p. 179
Augmentation Powers	p. 187
Augmentation Identities	p. 187
Integral Augmentation Powers	p. 187
Intersection Theorems	p. 190
Transfinite Augmentation Powers	p. 191
Schur Multiplicator	p. 196
Relative Dimension Subgroups	p. 210
A Characterization of Para-free Groups	p. 214
[tau]-para-free Groups	p. 215
Homological Localization of Z[G]-modules	p. 219
Homotopical Aspects	p. 229
The Associated Graded Ring of a Group Ring	p. 229
Spectral Sequences	p. 233
Applications to Dimension Subgroups	p. 243
Homotopical Applications	p. 254
Computations and Connectivity Results	p. 272
Miscellanea	p. 291
Power-closed Groups	p. 291
Braid Groups	p. 291
3-dimensional Surgery	p. 293
Vanishing Sums of Roots of Unity	p. 295
Fundamental Groups of Projective Curves	p. 296
Simplicial Methods	p. 299
Chain Complexes	p. 299
Simplicial Objects	p. 300
Geometric Realization Functor	p. 302
Skeleton and Coskeleton Functors	p. 302
Moore Complex and Homotopy Groups	p. 304
Dold-Kan Correspondence	p. 306
Eilenberg-Zilber Equivalence	p. 307
Classifying Functor W and Homology	p. 308
Bisimplicial Groups	p. 309
Certain Simplicial Constructions	p. 310
Free Simplicial Resolutions	p. 314
Functorial Properties	p. 317
Derived Functors	p. 320
Quadratic Functors	p. 322
Derived Functors in the Sense of Dold and Puppe	p. 326
Derived Limits and Fibration Sequence	p. 328
References	p. 331
Index	p. 343
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Mathematics

At a Glance

Paperback

Industry Reviews

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Defective items

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