Group and ring theoretic properties of polycyclic groups.
type: Item
instance of: scholarly article
title: Group and ring theoretic properties of polycyclic groups.
zbMATH Open document ID: 1206.20042
DOI: 10.1007/978-1-84882-941-1
published in: Algebra and Applications
publication date: 2009-11-17T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16-02
zbMATH DE Number: 5633959
zbMATH Keywords: polycyclic groups
zbMATH Keywords: soluble groups
zbMATH Keywords: polycyclic-by-finite groups
zbMATH Keywords: group theoretic properties
zbMATH Keywords: ring theoretic properties
author: B. A. F. Wehrfritz
reviewed by: Bùi Xuân Hải
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-84882-941-1
OpenAlex ID: W646298705
description: scientific article; zbMATH DE number 5633959