Infinite Abelian groups

375 Pages·1973·20.59 MB·English

by  Laszlo Fuchs

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.
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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