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A comprehensive guide to ranks and group theory Ranks of Groups features a logical, straightforward presentation, beginning with a succinct discussion of the standard ranks before moving on to specific aspects of ranks of groups. Topics covered include section ranks, groups of finite 0-rank, minimax rank, special rank, groups of finite section p-rank, groups having finite section p-rank for all primes p, groups of finite bounded section rank, groups whose abelian subgroups have finite rank, groups whose abelian subgroups have bounded finite rank, finitely generated groups having finite rank, residual properties of groups of finite rank, groups covered by normal subgroups of bounded finite rank, and theorems of Schur and Baer. This book presents fundamental concepts and notions related to the area of ranks in groups. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, this book focuses on critical concepts with the most interesting, striking, and central results. In order to provide readers with the most useful techniques related to the various different ranks in a group, the authors have carefully examined hundreds of current research articles on group theory authored by researchers around the world, providing an up-to-date, comprehensive treatment of the subject. * All material has been thoroughly vetted and class-tested by well-known researchers who have worked in the area of rank conditions in groups * Topical coverage reflects the most modern, up-to-date research on ranks of groups * Features a unified point-of-view on the most important results in ranks obtained using various methods so as to illustrate the role those ranks play within group theory * Focuses on the tools and methods concerning ranks necessary to achieve significant progress in the study and clarification of the structure of groups Ranks of Groups: The Tools, Characteristics, and Restrictions is an excellent textbook for graduate courses in mathematics, featuring numerous exercises, whose solutions are provided. This book will be an indispensable resource for mathematicians and researchers specializing in group theory and abstract algebra. MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama. LEONID A. KURDACHENKO, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. IGOR YA SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.

Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama.Leonid A. Kurdachenko, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.Igor Ya Subbotin, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.

*Ranks of Groups* features a logical, straightforward presentation, beginning with a succinct discussion of the standard ranks before moving on to specific aspects of ranks of groups. Topics covered include section ranks, groups of finite 0-rank, minimax rank, special rank, groups of finite section p-rank, groups having finite section p-rank for all primes p, groups of finite bounded section rank, groups whose abelian subgroups have finite rank, groups whose abelian subgroups have bounded finite rank, finitely generated groups having finite rank, residual properties of groups of finite rank, groups covered by normal subgroups of bounded finite rank, and theorems of Schur and Baer.

This book presents fundamental concepts and notions related to the area of ranks in groups. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, this book focuses on critical concepts with the most interesting, striking, and central results. In order to provide readers with the most useful techniques related to the various different ranks in a group, the authors have carefully examined hundreds of current research articles on group theory authored by researchers around the world, providing an up-to-date, comprehensive treatment of the subject.

- All material has been thoroughly vetted and class-tested by well-known researchers who have worked in the area of rank conditions in groups
- Topical coverage reflects the most modern, up-to-date research on ranks of groups
- Features a unified point-of-view on the most important results in ranks obtained using various methods so as to illustrate the role those ranks play within group theory
- Focuses on the tools and methods concerning ranks necessary to achieve significant progress in the study and clarification of the structure of groups

*Ranks of Groups: The Tools, Characteristics, and Restrictions* is an excellent textbook for graduate courses in mathematics, featuring numerous exercises, whose solutions are provided. This book will be an indispensable resource for mathematicians and researchers specializing in group theory and abstract algebra.

PrefaceChapter 1. Essential Toolbox 11.1. Ascending and Descending Series in Groups 11.2. Generalized Soluble Groups 71.3. Chernikov Groups and the Minimum Condition 191.4. Linear Groups 251.5. Some Relationships Between the Factors of the Upper and Lower Central Series 311.6. Some Direct Decompositions in Abelian Normal Subgroups 40Chapter 2. Groups of Finite 0-Rank 462.1. The Z-Rank in Abelian Groups 472.2. The 0-Rank of a Group 512.3. Locally Nilpotent Groups of Finite 0-Rank 532.4. Groups of Finite 0-Rank in General 572.5. Local Properties of Groups of Finite 0-Rank 64Chapter 3. Section p-Rank of Groups 713.1. p-Rank in Abelian Groups 713.2. Finite Section p-Rank 733.3. Locally Finite Groups with Finite Section p-Rank 853.4. Structure of Locally Generalized Radical Groups with Finite Section p-Rank 95Chapter 4. Groups of Finite Section Rank 984.1. Locally Finite Groups with Finite Section Rank 984.2. Structure of Locally Generalized Radical Groups with Finite Section Rank 1054.3. Connections Between the Order of a Finite Group and Its Section Rank 1104.4. Groups of Finite Bounded Section Rank 115Chapter 5. Zaitsev Rank 1215.1. The Zaitsev Rank of a Group 1215.2. Zaitsev Rank and 0-Rank 1275.3. Weak Minimal and Weak Maximal Conditions 131Chapter 6. Special Rank 1356.1. Elementary Properties of Special Rank 1356.2. The Structure of Groups Having Finite Special Rank 1416.3. The Relationship Between the Special Rank and the Bounded Section Rank 1526.4. A Taste of the Exotic 160Chapter 7. The Relationship Between the Factors of the Upper Central Series and the Nilpotent Residual 1647.1. Hypercentral Extensions by Groups of Finite 0-Rank 1647.2. Central Extensions by Groups of Finite Section Rank 1787.3. Hypercentral Extensions by Groups of Finite Section p-Rank 191Chapter 8. Finitely Generated Groups of Finite Section Rank 2058.1. The Z(G)-Decomposition in Some Abelian Normal Subgroups 2058.2. Splittings over Some Normal Subgroups 2148.3. Residually Finite Groups Having Finite 0-Rank 2228.4. Supplements to Divisible Abelian Normal Subgroups 228Chapter 9. The Influence of Important Families of Subgroups of Finite Rank 2409.1. The Existence of Supplements to the Hirsch-Plotkin Radical 2419.2. Groups Whose Locally Minimax Subgroups Have Finite Rank 2479.3. Groups Whose Abelian Subgroups Have Finite Rank 256Chapter 10. A Brief Discussion of Other Interesting Results 26110.1. Recent Work 26110.2. Questions 272Bibliography 276Author Index 295Symbol Index 297Subject Index 300

- Titel
- Ranks of Groups

- Untertitel
- The Tools, Characteristics, and Restrictions

- Autor

- EAN
- 9781119080329

- ISBN
- 978-1-119-08032-9

- Format
- E-Book (pdf)

- Hersteller
- Wiley

- Herausgeber
- Wiley

- Genre
- Arithmetik, Algebra

- Veröffentlichung
- 15.06.2017

- Digitaler Kopierschutz
- Adobe-DRM

- Dateigrösse
- 3.32 MB

- Anzahl Seiten
- 328

- Jahr
- 2017