INFINITE ABELIAN GROUPS Volume II This is Volume 36-11 in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: PAULA . SMITHA ND SAMUEL EILENBERC A complete list of titles in this series appears at the end of this volume INFINITE ABELIAN GROUPS LcEszld Fuchs Tulane University New Orleans, Louisiana V O L U M E I1 ACADEMIC PRESS NEW YORK AND LONDON 1973 A Subsidiary of Harmurt Brace Jovanovich, Publishers COPYRIGHT @ 1973, BY ACADEMIPCR ESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. 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 NWl LIBRARYOF CONQRE~S CATALOCGA RDN UMBEX 78-97479 AMS (MOS) 1970 Sub;ect Classification: 20~102,0 K15, 20K20, 20K30,20335. PRINTED IN THE UNITED STATES OF AMERICA 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 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 material on which research in abelian groups can be based. The treatment is by no means intended to be exhaustive or V vi 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 signi- ficant 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 II 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 Preface V XI. Separable p-Groups 65. Lemmas on p-Groups 1 66. Subsocles 6 67. Fully Invariant and Large Subgroups 9 68. Torsion-Complete Groups 14 69. Further Characterizations of Torsion-Complete p-Groups 21 70. Topological Completeness of Torsion-Complete Groups 24 71. Direct Decompositions of Torsion-Complete Groups 30 72. The Exchange Property 32 73. Direct Sums of Torsion-Complete Groups 37 74. Quasi-Complete Groups 45 75. Direct Decompositions of p-Groups 50 Notes 54 XII. p-Groups with Elements of Infinite Height 76. Existence Theorems on p-Groups 57 77. Ulm's Theorem 60 78. Direct Sums of Countable p-Groups 61 79. Nice Subgroups 73 80. Isotype and Balanced Subgroups 75 81. p-Groups with Nice Composition Series 81 vii viii CONTENTS 82. Totally Projective p-Groups 89 83. Simply Presented p-Groups 94 84. Summable p-Groups 102 Notes 105 XIII. Torsion-Free Groups 85. Torsion-Free Groups of Rank 1 107 86. Completely Decomposable Groups 112 87. Separable Groups 117 88. Indecomposable Groups 122 89. Large Indecomposable Groups 129 90. Direct Decompositions of Finite Rank Groups 134 91. Direct Decompositions of Countable Groups 141 92. Quasi-Direct Decompositions 148 93. Countable Torsion-Free Groups 154 94. Slender Groups 158 95. Characterization of Slender Groups by Subgroups 164 96. Vector Groups 167 97. Finite-Valued Functions into a Group 172 98. Homogeneous and Homogeneously Decomposable Groups 176 99. Whitehead’s Problem 178 Notes 181 XIV. Mixed Groups 100. Splitting Mixed Groups 185 101. Baer Groups Are Free 189 102. Quasi-Splitting Mixed Groups 194 103. Height-Matrices 197 104. Mixed Groups of Torsion-Free Rank 1 203 105. Groups with Prescribed Ulm Sequences 207 Notes 213 XV. Endomorphism Rings 106. Endomorphism Rings 21 5 107. Topologies of Endomorphism Rings 22 1 108. Endomorphism Rings of Torsion Groups 224 109. Endomorphism Rings of Separable p-Groups 228 110. Countable Torsion-Free Endomorphism Rings 231 11I . Endomorphism Rings with Special Properties 235 112. Regular and Generalized Regular Endomorphism Rings 239 Notes 246 XVI. Automorphism Groups 1 13. Groups of Automorphisms 249 114. Normal Subgroups in Automorphism Groups 255 CONTENTS ix 115. Automorphism Groups of Torsion Groups 262 116. Automorphism Groups of Torsion-Free Groups 268 Notes 275 XVII. Additive Groups of Rings 117. Subgroups That Are Always Ideals 277 118. Multiplications on a Group 280 119. Extensions of Partial Multiplications 284 120. Torsion Rings 287 121. Torsion-Free Rings 291 122. Additive Groups of Artinian Rings 295 123. Artinian Rings without Quasicyclic Subgroups 299 124. Additive Groups of Regular and n-Regular Rings 302 125. Embeddings in Regular and x-Regular Rings with Identity 305 126. Additive Groups of Noetherian Rings and Rings with Restricted Minimum Condition 308 Notes 31 1 XVIII. Groups of Units in Rings 127. Multiplicative Groups of Fields 312 128. Units of Commutative Rings 317 129. Groups That Are Unit Groups 320 Notes 324 Bibliography 325 Table of Notations 353 357 Author Index 361 Subject Index