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E-grāmata: Infinite Groups: A Roadmap to Selected Classical Areas

Leonid A. Kurdachenko, Igor Ya. Subbotin, Martyn R. Dixon

Formāts: 410 pages
Izdošanas datums: 26-Jan-2023
Izdevniecība: Chapman & Hall/CRC
Valoda: eng
ISBN-13: 9781000848311

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Formāts: 410 pages
Izdošanas datums: 26-Jan-2023
Izdevniecība: Chapman & Hall/CRC
Valoda: eng
ISBN-13: 9781000848311

Citas grāmatas par šo tēmu:

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In recent times, group theory has found wider applications in various fields of algebra and mathematics in general. But in order to apply this or that result, you need to know about it, and such results are often diffuse and difficult to locate, necessitating that readers construct an extended search through multiple monographs, articles, and papers. Such readers must wade through the morass of concepts and auxiliary statements that are needed to understand the desired results, while it is initially unclear which of them are really needed and which ones can be dispensed with. A further difficulty that one may encounter might be concerned with the form or language in which a given result is presented. For example, if someone knows the basics of group theory, but does not know the theory of representations, and a group theoretical result is formulated in the language of representation theory, then that person is faced with the problem of translating this result into the language with which they are familiar, etc.

Infinite Groups: A Roadmap to Selected Classical Areas

seeks to overcome this challenge. The book covers a broad swath of the theory of infinite groups, without giving proofs, but with all the concepts and auxiliary results necessary for understanding such results. In other words, this book is an extended directory, or a guide, to some of the more established areas of infinite groups.

Features

An excellent resource for a subject formerly lacking an accessible and in-depth reference

Suitable for graduate students, PhD students, and researchers working in group theory

Introduces the reader to the most important methods, ideas, approaches, and constructions in infinite group theory.

This book covers a broad swath of the theory of infinite groups, without giving proofs, but with all the concepts and auxiliary results necessary for understanding such results. In other words, this book is an extended directory, or a guide, to some of the more established areas of infinite groups.

Preface

Authors

xvii

Chapter 1 Important Subgroups

(48)

1.1 Some Important Series In Groups And Subgroups Defined By These Series

(8)

1.2 Classes Of Groups Defined By Series Of Subgroups

(6)

1.3 Radicable Groups

(3)

1.4 Something From The Theory Of Modules

(1)

1.5 The 0-Rank And P-Rank Of Abelian Groups

(3)

1.6 The Frattini Subgroup Of A Group

(3)

1.7 Linear Groups

(5)

1.8 Residually X-Groups

(16)

References for
Chapter 1

(8)

Chapter 2 Finitely Generated Groups

(34)

2.1 The Generalized Burnside Problem

(1)

2.2 The Burnside Problem For Groups Of Finite Exponent

(2)

2.3 The Restricted Burnside Problem

(1)

2.4 Growth Functions On Finitely Generated Groups

(3)

2.5 Finitely Presented Groups

(3)

2.6 Groups With The Maximal Condition For All Subgroups

(22)

References for
Chapter 2

(11)

Chapter 3 Finiteness Conditions

(44)

3.1 The Minimal Condition On Certain Systems Of Subgroups

(7)

3.2 The Minimal Condition On Normal Subgroups

(3)

3.3 Artinian And Related Modules Over Some Group Rings

(9)

3.4 Minimax Groups

102

(5)

3.5 The Weak Minimal Condition

107

(7)

3.6 The Weak Maximal Condition

114

(13)

References for
Chapter 3

117

(10)

Chapter 4 Ranks of Groups

127

(32)

4.1 Finite Special Rank And Finite Section P-Rank

127

(3)

4.2 Finite 0-Rank

130

(3)

4.3 The Connections Between The Various Rank Conditions I

133

(1)

4.4 Finite Section Rank

134

(4)

4.5 Bounded Section Rank

138

(1)

4.6 The Connections Between The Various Rank Conditions II

139

(4)

4.7 Finitely Generated Groups

143

(3)

4.8 Systems Of Subgroups Satisfying Rank Conditions

146

(3)

4.9 Some Residual Systems

149

(10)

References for
Chapter 4

153

(6)

Chapter 5 Conjugacy Classes

159

(44)

5.1 Around "Schur's Theorem", Central-By-Finite Groups And Related Topics

160

(12)

5.2 Bounded Conjugacy Classes, Finite-By-Abelian Groups And Related Classes

172

(5)

5.3 Groups With Finite Classes Of Conjugate Elements

177

(15)

5.4 Some Concluding Remarks

192

(11)

References for
Chapter 5

193

(10)

Chapter 6 Generalized Normal Subgroups and their Opposites

203

(66)

6.1 Groups Whose Subgroups Are Normal, Permutable Or Subnormal

204

(7)

6.2 Groups Having A Large Family Of Normal Subgroups

211

(7)

6.3 Groups Having A Large Family Of Subnormal Subgroups

218

(7)

6.4 Pairs Of Opposite Subgroups

225

(9)

6.5 Transitively Normal Subgroups

234

(9)

6.6 The Norm Of A Group, The Wielandt Subgroup And Related Topics

243

(6)

6.7 The Norm Of A Group And The Quasicentralizer Condition

249

(20)

References for
Chapter 6

254

(15)

Chapter 7 Locally Finite Groups

269

(46)

7.1 Preliminaries

269

(3)

7.2 Large Locally Finite Groups

272

(3)

7.3 Simple Locally Finite Groups

275

(7)

7.4 Existentially Closed Groups

282

(2)

7.5 Centralizers In Locally Finite Groups

284

(4)

7.6 Sylow Theory In Locally Finite Groups

288

(2)

7.7 Conjugacy Of Sylow Subgroups

290

(4)

7.8 Unconventional Sylow Theories

294

(3)

7.9 Saturated Formations And Fitting Classes

297

(4)

7.10 Barely Transitive Groups

301

(14)

References for
Chapter 7

303

(12)

Bibliography

315

(62)

Author Index

377

(4)

Symbol Index

381

(4)

Subject Index

385

Dr. Martyn R. Dixon is a Professor of Mathematics at the University of Alabama. He did undergraduate work at the University of Manchester and obtained his Ph. D. at the University of Warwick under the guidance of Dr. Stewart Stonehewer. His main interests in group theory include ranks of groups, infinite dimensional linear groups, permutable subgroups and locally finite groups. He has written several books and numerous articles concerned with group theory. He has been a visiting professor at various institutions including the University of Kentucky, Bucknell University, Universitą degli Studi di Trento, the University of Napoli, the University of Salerno, the University of Valencia and the University of Zarogoza.

Dr. Leonid A. Kurdachenko

is a Distinguished Professor in the Department of Geometry and Algebra of Oles Honchar Dnipro National University. He is one of the most productive group theorists. His list of publications consists of more than 250 journal articles published in major mathematics journals in many countries around the globe. He is an author of more than a dozen books published by such prestigious publishers as John Wiley and Sons (USA), Birkhäuser (Swiss), Word Scientific (United Kingdom), and others. He served as an invited speaker and visiting professor at many international conferences and universities. His research activities have been supported by several prestigious international grants. Dr. Igor Ya. Subbotin

is a Professor at National University, USA. His main area of research is algebra. His list of publications includes more than 170 articles in algebra published in major mathematics journals around the globe. He has also authored more than 50 articles in mathematics education dedicated mostly to the theoretical basis of some topics in high school and college mathematics. Among his publications there are several books published by such major publishing companies as Wiley and Sons, World Scientific, Birkhäuser, and others. His research in algebra has been supported by several international prestigious grants, including grants issued by FEDER funds from European Union, The National Research Committee of Spain and Aragon, Volkswagen Foundation (VolkswagenStiftung), and others.

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