Foreword.- Some basic group theory.- Some ring theory.- Soluble linear groups.- Further group-theoretic properties of polycyclic groups.- Groups acting on finitely generated commutative rings.- Prime ideals in polycyclic-group rings.- The structure of modules over polycyclic groups.- Semilinear and skew linear groups.
Some Basic Group Theory.- The Basic Theory of Polycyclic Groups.- Some Ring Theory.- Soluble Linear Groups.- Further Group-Theoretic Properties of Polycyclic Groups.- Hypercentral Groups and Rings.- Groups Acting on Finitely Generated Commutative Rings.- Prime Ideals in Polycyclic Group Rings.- The Structure of Modules over Polycyclic Groups.- Semilinear and Skew Linear Groups.
Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in algebra, for example in the theory of Noetherian rings. The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and the basic properties of these groups. The second half then focuses specifically on the ring theoretic properties of polycyclic groups and their applications, often to purely group theoretic situations.
The book is intended to be a study manual for graduate students and researchers coming into contact with polycyclic groups, where the main lines of the subject can be learned from scratch. Thus it has been kept short and readable with a view that it can be read and worked through from cover to cover. At the end of each topic covered there is a description without proofs, but with full references, of further developments in the area. An extensive bibliography then concludes the book.
A short and concise treatment of the essential results with proofs that are clear and easy to follow. This book will prepare readers for research in related areas.
Accessible to researchers working in areas other than group theory who find themselves involved with polycyclic groups; no previous knowledge of polycyclic groups is assumed.
Introduces all the various techniques used in the proof of Roseblade's residual finiteness theorem.
Written by a renowned expert in the field of infinite groups.