Abelian Groups

Abelian Groups deals with the theory of abelian or commutative groups, with special emphasis on results concerning structure problems. More than 500 exercises of varying degrees of difficulty, with and without hints, are included. Some of the exercises illuminate the theorems cited in the text by providing alternative developments, proofs or counterexamples of generalizations. Comprised of 16 chapters, this volume begins with an overview of the basic facts on group theory such as factor group or homomorphism. The discussion then turns to direct sums of cyclic groups, divisible groups, and direct summands and pure subgroups, as well as Kulikovs basic subgroups. Subsequent chapters focus on the structure theory of the three main classes of abelian groups: the primary groups, the torsion-free groups, and the mixed groups. Applications of the theory are also considered, along with other topics such as homomorphism groups and endomorphism rings; the Schreier extension theory with a discussion of the group of extensions and the structure of the tensor product. In addition, the book examines the theory of the additive group of rings and the multiplicative group of fields, along with Baers theory of the lattice of subgroups. This book is intended for young research workers and students who intend to familiarize themselves with abelian groups. Content: Front Matter, Page 3 Copyright, Page 4 PREFACE, Pages 5-8 TABLE OF NOTATIONS, Page 12 CHAPTER I - BASIC CONCEPTS. THE MOST IMPORTANT GROUPS, Pages 13-36 CHAPTER II - DIRECT SUM OF CYCLIC GROUPS, Pages 37-56 CHAPTER III - DIVISIBLE GROUPS, Pages 57-70 CHAPTER IV - DIRECT SUMMANDS AND PURE SUBGROUPS, Pages 71-96 CHAPTER V - BASIC SUBGROUPS, Pages 97-110 CHAPTER VI - THE STRUCTURE OF p-GROUPS, Pages 111-144 CHAPTER VII - TORSION FREE GROUPS, Pages 145-184 CHAPTER VIII - MIXED GROUPS, Pages 185-204 CHAPTER IX - HOMOMORPHISM GROUPS AND ENDOMORPHISM RINGS, Pages 205-232 CHAPTER X - GROUP EXTENSIONS, Pages 233-248 CHAPTER XI - TENSOR PRODUCTS, Pages 249-257 CHAPTER XII - THE ADDITIVE GROUP OF RINGS, Pages 258-294 CHAPTER XIII - THE MULTIPLICATIVE GROUP OF FIELDS, Pages 295-299 CHAPTER XIV - THE LATTICE OF SUBGROUPS, Pages 300-314 CHAPTER XV - DECOMPOSITIONS INTO DIRECT SUMS OF SUBSETS, Pages 315-331 CHAPTER XVI - VARIOUS QUESTIONS, Pages 332-352 BIBLIOGRAPHY, Pages 353-361 AUTHOR INDEX, Pages 363-364 SUBJECT INDEX, Pages 365-367 ERRATA, Page ibc1