# Ranks of Groups

## The Tools, Characteristics, and Restrictions

1. Edition September 2017

328 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-08027-5**

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 9

1.4 Linear Groups 5

1.5 Some Relationships Between the Factors of the Upper and Lower Central Series 1

1.6 Some Direct Decompositions in Abelian Normal Subgroups 0

Chapter 2 Groups of Finite 0-Rank 6

2.1 The Z-Rank in Abelian Groups 7

2.2 The 0-Rank of a Group 1

2.3 Locally Nilpotent Groups of Finite 0-Rank 3

2.4 Groups of Finite 0-Rank in General 7

2.5 Local Properties of Groups of Finite 0-Rank 4

Chapter 3 Section p-Rank of Groups 1

3.1 p-Rank in Abelian Groups 1

3.2 Finite Section p-Rank 3

3.3 Locally Finite Groups with Finite Section p-Rank 5

3.4 Structure of Locally Generalized Radical Groups with Finite Section p-Rank 5

Chapter 4 Groups of Finite Section Rank 8

4.1 Locally Finite Groups with Finite Section Rank 8

4.2 Structure of Locally Generalized Radical Groups with Finite Section Rank 1 5

4.3 Connections Between the Order of a Finite Group and Its Section Rank 1 0

4.4 Groups of Finite Bounded Section Rank 1 5

Chapter 5 Zaitsev Rank 1 1

5.1 The Zaitsev Rank of a Group 1 1

5.2 Zaitsev Rank and 0-Rank 1 7

5.3 Weak Minimal and Weak Maximal Conditions 1 1

Chapter 6 Special Rank 1 5

6.1 Elementary Properties of Special Rank 1 5

6.2 The Structure of Groups Having Finite Special Rank 1 1

6.3 The Relationship Between the Special Rank and the Bounded Section Rank 1 2

6.4 A Taste of the Exotic 1 0

Chapter 7 The Relationship Between the Factors of the Upper Central Series and the Nilpotent Residual 1 4

7.1 Hypercentral Extensions by Groups of Finite 0-Rank 1 4

7.2 Central Extensions by Groups of Finite Section Rank 1 8

7.3 Hypercentral Extensions by Groups of Finite Section p-Rank 1

Chapter 8 Finitely Generated Groups of Finite Section Rank 2 5

8.1 The Z(G)-Decomposition in Some Abelian Normal Subgroups 2 5

8.2 Splittings over Some Normal Subgroups 2 4

8.3 Residually Finite Groups Having Finite 0-Rank 2 2

8.4 Supplements to Divisible Abelian Normal Subgroups 2 8

Chapter 9 The Inuence of Important Families of Subgroups of Finite Rank 2 0

9.1 The Existence of Supplements to the Hirsch{Plotkin Radical 2 1

9.2 Groups Whose Locally Minimax Subgroups Have Finite Rank 2 7

9.3 Groups Whose Abelian Subgroups Have Finite Rank 2 6

Chapter 10 A Brief Discussion of Other Interesting Results 2 1

10.1 Recent Work 2 1

10.2 Questions 2 2

Bibliography 2 6

Author Index 5

Symbol Index 7

Subject Index 3

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.