Infinite Abelian Groups

305 Pages·1970·9.93 MB·English

by  Laszlo Fuchs

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
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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
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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