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Theory of Groups, Volume 1: Second English Edition
A. G. Kurosh

AMS Chelsea Publishing
1955; 272 pp; hardcover
Reprint/Revision History:
reprinted 1960
ISBN-10: 0-8284-0107-1
ISBN-13: 978-0-8284-0107-4
List Price: US$47
Member Price: US$35.58
Order Code: CHEL/107
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Theory of Groups, Volume 2 - A G Kurosh

Translated from the second Russian edition and with added notes by K. A. Hirsch. Teoriya Grupp by Kurosh was widely acclaimed, in its first edition, as the first modern text on the general theory of groups, with the major emphasis on infinite groups. The decade that followed brought about a remarkable growth and maturity in the theory of groups, so that this second edition, an English translation, represents a complete rewriting of the first edition. The book can be used as a beginning text, the only requirement being some mathematical maturity and a knowledge of the elements of transfinite numbers.

Many new sections were added to this second edition, and many old ones were completely revised: The theory of abelian groups was significantly revised; many significant additions were made to the section on the theory of free groups and free products; an entire chapter is devoted to group extensions; and the deep changes in the theory of solvable and nilpotent groups--one of the large and rich branches of the theory of groups--are covered in this work. Each volume concludes with Editor's Notes and a Bibliography.

Table of Contents

Part One. The Elements of Group Theory
  • Definition of a Group: 1.1 Algebraic operations; 1.2 Isomorphism. Homomorphism; 1.3 Groups; 1.4 Examples of groups
  • Subgroups: 2.5 Subgroups; 2.6 Systems of generators. Cyclic groups; 2.7 Ascending sequences of groups
  • Normal Subgroups: 3.8 Decomposition of a group with respect to a subgroup; 3.9 Normal subgroups; 3.10 The connection between normal subgroups, homomorphisms, and factor groups; 3.11 Classes of conjugate elements, and conjugate subgroups
  • Endomorphisms and Automorphisms. Groups with Operators: 4.12 Endomorphisms and automorphisms; 4.13 The holomorph. Complete groups; 4.14 Characteristic and fully invariant subgroups; 4.15 Groups with operators
  • Series of Subgroups. Direct Products. Defining Relations: 5.16 Normal series and composition series; 5.17 Direct products; 5.18 Free groups. Defining relations
Part Two. Abelian Groups
  • Foundations of the theory of abelian groups; 6.19 The rank of an abelian group. Free abelian groups; 6.20 Finitely generated abelian groups; 6.21 The ring of endomorphisms of an abelian group; 6.22 Abelian groups with operators
  • Primary and Mixed Abelian Groups: 7.23 Complete abelian groups; 7.24 Direct sums of cyclic groups; 7.25 Serving subgroups; 7.26 Primary groups without elements of infinite height; 7.27 Ulm factors. The existence theorem; 7.28 Ulm's theorem; 7.29 Mixed abelian groups
  • Torsion-Free Abelian Groups: 8.30 Groups of rank 1. Types of elements of torsion-free groups; 8.31 Completely decomposable groups; 8.32 Other classes of abelian torsion-free groups
  • Appendixes
  • Bibliography
  • Author Index
  • Subject Index
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