1st Edition

# Infinite Groups A Roadmap to Selected Classical Areas

410 Pages
by Chapman & Hall

410 Pages
by Chapman & Hall

Also available as eBook on:

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.

1. Important Subgroups. 1.1. Some Important Series in Groups and Subgroups Defined by these Series. 1.2. Classes of Groups Defined by Series of Subgroups. 1.3. Radicable Groups. 1.4. Something from the Theory of Modules. 1.5. The 0-Rank and p-rank of Abelian Groups. 1.6. The Frattini Subgroup of a Group. 1.7. Linear Groups. 1.8. Residually X-Groups. 2. Finitely Generated Groups. 2.1. The Generalized Burnside Problem. 2.2. The Burnside Problem for Groups of Finite Exponent. 2.3. The Restricted Burnside Problem. 2.4. Growth Functions on Finitely Generated Groups. 2.5. Finitely Presented Groups. 2.6. Groups with the Maximal Condition for all Subgroups. 3. Finiteness Conditions. 3.1 The Minimal Condition on Certain Systems of Subgroups. 3.2. The Minimal Condition on Normal Subgroups. 3.3. Artinian and Related Modules over some Group Rings. 3.4. Minimax Groups. 3.5. The Weak Minimal Condition. 3.6. The Weak Maximal Condition. 4. Ranks of Groups. 4.1. Finite Special Rank and Finite Section p-Rank. 4.2. Finite 0-Rank. 4.3. The Connections Between the Various Rank Conditions I. 4.4. Finite Section Rank. 4.5. Bounded Section Rank. 4.6. The Connections Between the Various Rank Conditions II. 4.7. Finitely Generated Groups. 4.8. Systems of Subgroups Satisfying Rank Conditions. 4.9. Some Residual Systems. 5. Conjugacy Classes. 5.1. Around "Schur’s Theorem", Central-By-Finite Groups and Related Topics. 5.2 Bounded Conjugacy Classes, Finite-By-Abelian Groups and Related Classes. 5.3. Groups with Finite Classes of Conjugate Elements. 5.4. Some Concluding Remarks. 6. Generalized Normal Subgroups and their Opposites. 6.1. Groups Whose Subgroups are Normal, Permutable or Subnormal. 6.2. Groups having a Large Family of Normal Subgroups. 6.3. Groups having a Large Family of Subnormal Subgroups. 6.4. Pairs of Opposite Subgroups. 6.5. Transitively Normal Subgroups. 6.6. The Norm of a Group, The Wielandt Subgroup and Related Topics. 6.7. The Norm of a Group and the Quasicentralizer Condition. 7. Locally Finite Groups. 7.1. Preliminaries. 7.2. Large Locally Finite Groups. 7.3. Simple Locally Finite Groups. 7.4. Existentially Closed Groups. 7.5. Centralizers in Locally Finite Groups. 7.6. Sylow Theory in Locally Finite Groups. 7.7. Conjugacy of Sylow Subgroups. 7.8. Unconventional Sylow Theories. 7.9. Saturated Formations and Fitting Classes. 7.10. Barely Transitive Groups.

### Biography

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, Università 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.