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Infinite Abelian Groups, Volume 1 (Pure & Applied Mathematics) PDF

305 Pages·1970·1.76 MB·English
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INFINITE ABELIAN GROUPS Volume Z This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEELI LENBERANGD HYMABNA SS A list of recent titles in this series appears at the end of this volume INFINITE ABELIAN GROUPS Ld.szlo Fuchs Ttclane University New Orleans, Louisiana V O L U M E I ACADEMIC PRESS New Yorh Sail Frawiscv Loizdon 1970 A Subsidiary of Harcoitrt Brace Jovanovich, Publishers COPYRIGH@T 1970, BY ACADEMIPCR ESSJ,N C. ALL RIGHTS RESERVED NO PART OF THlS ROOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC, PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWI LIBRAROYF CONGRECSSA TALOCGA RDN UMBER78: -97479 AMS 1968 SUBJECT CLASSIFICAT2I0O3N0 PRINTED IN THE UNITED STATES OF AMERICA 79808182 9 8 7 6 S 4 3 Patri meo DAVID RAPHAEL FOKOS-FUCHS doctissimo magistro linguarum Fenno- Ugricarum summo cum amore pi0 ac gratioso This Page Intentionally Left Blank PREFACE The theory of abelian groups is a branch of algebra which deals with commutative groups. Curiously enough, it is rather independent of general group theory: its basic ideas and methods bear only a slight resemblance to the noncommutative case, and there are reasons to believe that no other condition on groups is more decisive for the group structure than com- mutativity. The present book is devoted to the theory of abelian groups. The study of abelian groups may be recommended for two principal reasons: in the first place, because of the beauty of the results which include some of the best examples of what is called algebraic structure theory; in the second place, it is one of the principal motives of new research in module theory (e.g., for every particular theorem on abelian groups one can ask over what rings the same result holds) and there are other areas of mathematics in which exten- sive use of abelian group theory might be very fruitful (structure of homology groups, etc.). It was the author’s original intention to write a second edition of his book “Abelian Groups” (Budapest, 1958). However, it soon became evident that in the last decade the theory of abelian groups has moved too rapidly for a mere revised edition, and consequently, a completely new book has been written which reflects the new aspects of the theory. Some topics (lattice of subgroups, direct decompositions into subsets, etc.) which were treated in “Abelian Groups” will not be touched upon here. The twin aims of this book are to introduce graduate students to the theory of abelian groups and to provide a young algebraist with a reasonably comprehensive summary of the matenu. >nw hich research in abelian groups can be based. The treatment is by no means intended to be exhaustive or vii ... Vlll PREFACE even to yield a complete record of the present status of the theory-this would have been a Sisyphean task, since the subject has become so extensive and is growing almost from day to day. But the author has tried to be fairly com- plete in what he considers as the main body of up-to-date abelian group theory, and the reader should get a considerable amount of knowledge of the central ideas, the basic results, and the fundamental methods. To assist the reader in this, numerous exercises accompany the text; some of them are straightforward, others serve as additional theory or contain various com- plements. The exercises are not used in the text except for other exercises, but the reader is advised to attempt some exercises to get a better under- standing of the theory. No mathematical knowledge is presupposed beyond the rudiments of abstract algebra, set theory, and topology; however, a certain maturity in mathematical reasoning is required. The selection of material is unavoidably somewhat subjective. The main emphasis is on structural problems, and proper place is given to homological questions and to some topological considerations. A serious attempt has been made to unify methods, to simplify presentation, and to make the treatment as self-contained as possible. The author has tried to avoid making the discussion too abstract or too technical. With this view in mind, some significant results could not be treated here and maximum generality has not been achieved in those places where this would entail a loss of clarity or a lot of technicalities. Volume I presents what is fundamental in abelian groups together with the homological aspects of the theory, while Volume I1 is devoted to the structure theory and to applications. Each volume has a Bibliography listing those works on abelian groups which are referred to in the text. The author has tried to give credit wherever it belongs. In some instances, however, especially in the exercises, it was nearly impossible to credit ideas to their original discoverers. At the end of each chapter, some comments are made on the topics of the chapter, and some further results and generalizations (also to modules) are mentioned which a reader may wish to pursue. Also, research problems are listed which the author thought interesting. The system of cross-references is self-explanatory. The end of a proof is marked with the symbol 0.P roblems which, for some reason or other, seemed to be difficult are often marked by an asterisk, as are some sections which a beginning reader may find it wise to skip. The author is indebted to a number of group theorists for comments and criticisms; sincere thanks are due to all of them. Special thanks go to B. Charles for his numerous helpful comments. The author would like to express his gratitude to the Mathematics Departments of University of Miami, Coral Gables, Florida, and Tulane University, New Orleans, Louisiana, for their assistance in the preparation of the manuscript, and to Academic Press, Inc., for the publication of this book in their prestigious series. CONTENTS vii Preface I. Preliminaries 1. Definitions 1 2. Maps and Diagrams 6 3. The Most Important Types of Groups 14 4. Modules 19 5. Categories of Abelian Groups 21 6. Functorial Subgroups and Quotient Groups 25 7. Topologies in Groups 29 Notes 34 11. Direct Sums 8. Direct Sums and Direct Products 36 9. Direct Summands 46 10. Pullback and Pushout Diagrams 51 11, Direct Limits 53 12. Inverse Limits 59 13. Completeness and Completions 65 70 Notes ix

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