Ranks of Groups: The Tools, Characteristics, and Restrictions


thumbnail image: Ranks of Groups: The Tools, Characteristics, and Restrictions
  • Published: 15 September 2017
  • ISBN: 9781119080275
  • Author(s): Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin
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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.


Chapter 1. Essential Toolbox 1

1.1. Ascending and Descending Series in Groups 1

1.2. Generalized Soluble Groups 7

1.3. Chernikov Groups and the Minimum Condition 19

1.4. Linear Groups 25

1.5. Some Relationships Between the Factors of the Upper and Lower Central Series 31

1.6. Some Direct Decompositions in Abelian Normal Subgroups 40

Chapter 2. Groups of Finite 0-Rank 46

2.1. The Z-Rank in Abelian Groups 47

2.2. The 0-Rank of a Group 51

2.3. Locally Nilpotent Groups of Finite 0-Rank 53

2.4. Groups of Finite 0-Rank in General 57

2.5. Local Properties of Groups of Finite 0-Rank 64

Chapter 3. Section p-Rank of Groups 71

3.1. p-Rank in Abelian Groups 71

3.2. Finite Section p-Rank 73

3.3. Locally Finite Groups with Finite Section p-Rank 85

3.4. Structure of Locally Generalized Radical Groups with Finite Section p-Rank 95

Chapter 4. Groups of Finite Section Rank 98

4.1. Locally Finite Groups with Finite Section Rank 98

4.2. Structure of Locally Generalized Radical Groups with Finite Section Rank 105

4.3. Connections Between the Order of a Finite Group and Its Section Rank 110

4.4. Groups of Finite Bounded Section Rank 115

Chapter 5. Zaitsev Rank 121

5.1. The Zaitsev Rank of a Group 121

5.2. Zaitsev Rank and 0-Rank 127

5.3. Weak Minimal and Weak Maximal Conditions 131

Chapter 6. Special Rank 135

6.1. Elementary Properties of Special Rank 135

6.2. The Structure of Groups Having Finite Special Rank 141

6.3. The Relationship Between the Special Rank and the Bounded Section Rank 152

6.4. A Taste of the Exotic 160

Chapter 7. The Relationship Between the Factors of the Upper Central Series and the Nilpotent Residual 164

7.1. Hypercentral Extensions by Groups of Finite 0-Rank 164

7.2. Central Extensions by Groups of Finite Section Rank 178

7.3. Hypercentral Extensions by Groups of Finite Section p-Rank 191

Chapter 8. Finitely Generated Groups of Finite Section Rank 205

8.1. The Z(G)-Decomposition in Some Abelian Normal Subgroups 205

8.2. Splittings over Some Normal Subgroups 214

8.3. Residually Finite Groups Having Finite 0-Rank 222

8.4. Supplements to Divisible Abelian Normal Subgroups 228

Chapter 9. The Influence of Important Families of Subgroups of Finite Rank 240

9.1. The Existence of Supplements to the Hirsch-Plotkin Radical 241

9.2. Groups Whose Locally Minimax Subgroups Have Finite Rank 247

9.3. Groups Whose Abelian Subgroups Have Finite Rank 256

Chapter 10. A Brief Discussion of Other Interesting Results 261

10.1. Recent Work 261

10.2. Questions 272

Bibliography 276

Author Index 295

Symbol Index 297

Subject Index 300

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