Group and ring theoretic properties of polycyclic groups.
A group \(G\) is polycyclic if \(G\) has a series of subgroups \(\langle 1\rangle=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_n=G\), where \(n\) is an integer and \(G_{i+1}/G_i\) is cyclic for each \(i=0,1,\dots,n-1\). By a classical result (Mal'cev 1951), every soluble subgroup of \(\text{GL}(n,\mathbb{Z})\) is polycyclic. Conversely (Auslander 1967), every polycyclic-by-finite group is embedded in \(\text{GL}(n,\mathbb{Z})\) for some integer \(n\). Thus, polycyclic groups are exactly the soluble groups with faithful matrix representations over the integers. Another more important property of polycyclic groups is that they are exactly the soluble groups satisfying the maximal condition on subgroups. These basic properties of polycyclic groups make them to be important and attract the attention of a number of mathematicians. In fact, the polycyclic groups appear in many works, not only in the domain of the theory of groups: they appear also in works in other areas of research such as in ring theory, topology, \dots. The book under review consists of 10 chapters and is devoted to the systematic study of polycyclic groups from the beginning in the late 1930's up to now. The author makes attempts to describe more important results obtained and more powerful collections of techniques in the theory of polycyclic groups. The concept of polycyclic groups was introduced by Kurt Hirsch in the late 1930's. The basic theory of these groups is presented in Chapter 2 of the book. The situation essentially changed, when in the 1950's Mal'cev introduced the application of matrix group theory to the study of polycyclic groups. This area of research continued for some two decades and is reflected in Chapters 4 and 5. The remaining chapters are devoted to the very important period of the development of the theory that came from the 1970's, related with the ring theoretic methods. The ring theoretic properties of polycyclic groups were investigated and developed by a number of mathematicians, with J. E. Roseblade being a major contributor. These investigations are main objects presented in Chapters 3, 6, 7, 8 and 9. Many results show how to use the techniques of ring theory to the study of problems in group theory. Chapter 10 is devoted especially to the author's investigations that are concentrated on this ring theoretic area. The book is written clearly, with a high scientific level. As the author himself has mentioned in the Preface, the book is short and readable from start to finish. It is quite accessible to research workers not only in the area of group theory, but also in other areas, who find themselves involved with polycyclic groups. The Bibliography is rich and reflects the development of the theory from very early time up to now.
scientific article; zbMATH DE number 5633959