Exercises in Abelian Group Theory

353 Pages·2003·11.63 MB·English

by  Grigore Călugăreanu

ExercisesinAbelianGroupTheory
KluwerTexts inthe Mathematical Sciences
VOLUME25
AGraduate-LevelBookSeries
Thetitlespublishedinthisseriesarelistedattheendofthisvolume.
Exercises in Abelian
Group Theory
by
Grigore Calugareanu
'BabesBolyai'University,
Cluj-Napoca,RomaniaandKuwaitUniversity,StateofKuwait
Simion Breaz
'BabesBolyai'University,
Cluj-Napoca,Romania
Ciprian Modoi
'BabesBolyai'University,
Cluj-Napoca,Romania
Cosmin Pelea
'BabesBolyai'University,
Cluj-Napoca,Romania
and
DumitruValcan
'BabesBolyai'University,
Cluj-Napoca,Romania
Springer-Science+BusinessMedia,B.V.
Ac.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN 978-90-481-6249-9 ISBN 978-94-017-0339-0 (eBook)
DOI 10.1007/978-94-017-0339-0
Printedonacid-freepaper
AllRightsReserved
©2003SpringerScience+BusinessMediaDordrecht
OriginallypublishedbyKluwerAcademicPublishersin2003.
Softcoverreprintofthehardcover1stedition2003
Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmitted
inanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording
orotherwise,withoutwrittenpermissionfromthePublisher,withtheexception
ofanymaterialsuppliedspecificallyforthepurposeofbeingentered
andexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.
This volume is dedicated to the memory of our professor
Gheorghe Pic 1907- 1984
Contents
Preface vii
List of Symbols ix
I STATEMENTS 1
1 Basic notions.
Direct sums 3
2 Divisible groups 21
3 Pure subgroups.
Basic subgroups 29
4 Topological groups.
Linear topologies 39
5 Algebraically compact groups 49
6 Homological methods 55
7 p-groups 71
8 Torsion-free groups 85
9 Mixed groups 95
10 Subgroup lattices ofgroups 105
v
VI CONTENTS
II SOLUTIONS 111
1 Basic notions.
Direct sums 113
2 Divisible groups 143
3 Pure subgroups.
Basic subgroups 163
4 Topological groups.
Linear topologies 191
5 Algebraically compact groups 211
6 Homological methods 217
7 p-groups 239
8 Torsion-free groups 261
9 Mixed groups 295
10 Subgroup lattices ofgroups 319
Bibliography 333
Index 345
Preface
This book, in some sense, began to be written by the first author in
1983,when optional lectures on Abelian groups were held at the Fac
ulty ofMathematics and Computer Science,'Babes-Bolyai' University
in Cluj-Napoca, Romania. From 1992,these lectures wereextended to
a twosemester electivecourseon abelian groups forundergraduatestu
dents, followed bya twosemester courseonthe sametopic forgraduate
students in Algebra. Allthe other authors attended these two years of
lectures and are now Assistants to the Chair of Algebra of this Fac
ulty. The first draft of this collection, including only exercisessolved
bystudents as home works, the last ten years, had 160pages.
We felt that there is a need for a book such as this one, because
it would provide a nice bridge between introductory Abelian Group
Theory and more advanced research problems.
The book InfiniteAbelianGroups, published byLaszloFuchsintwo
volumes1970and 1973willwithout doubt last as the most important
guide for abelian group theorists. Many exercises are selected from
this source but there are plenty ofother bibliographical items (seethe
Bibliography) whichwereused in order to makeup this collection. For
someof the problems stated, recent developments are also given.
Nevertheless, there are plenty of elementary results (the so called
'folklore')inAbelian Group Theory whichdonot appearin anywritten
material. It is alsoone purpose ofthis book to complete this gap.
All the groups considered in this book are abelian and we adopt
the additive notation. All the notations are explained in the List of
Symbols.
Each chapter beginswith severalpagesofdefinitions orshort intro
ductions to notions and terminology not defined in standard graduate
Vll
Vlli PREFACE
algebratextbooks (calledPreliminaries). There arealsoresults which
the reader may want to use forsomeadvanced solutions.
Each chapter contains three kind of exercises: simple exercisesde
noted S 1, S 2, ..., exercises ofmedium difficultydenoted M 1, M 2,
... and difficultexercises denoted D 1, D 2, ... Inthis waythe beginner
(e.g., the undergraduate student) may solvefirst the S part of each a
chapter's exercises, continue with the M part, and, if after these he
definitelylikesAbelian Group Theory, he may finishwith the D part!
Each Exercises sectionends with a list ofreferences (Literature),
providing the interested readers with further information on the sub
ject.
In order to encourage the reader to apply as much as possible
Abelian Group Theory machinery, some exercises have more than one
solution.
OurcollectionhelpsonetostudyAbelianGroupTheory,discovering
and using the standard results in almost all the topics related to this
theory. It isalsointended to giveanew (and maybenecessary)impetus
to this veryattractive part ofalgebra. It isalsoagoldmineofexamples
forspecialists interested in Module Theory (whichlately seem to have
forgotten Abelian Group Theory!).
Finally,let usmention that the collectionisaddressedto undergrad
uate and graduate students, but hopefully it willbe of use to experts
in the field. For undergraduate students our advise was already given:
to begin with the S part (the simple exercises) and continue everyone
as much as it is willingto. Graduate students in algebra may skip the
S part, but maybe onlyafter having read the S statements. They must
feel comfortable in the M part (the medium exercises) and arestrongly
advised to continue as much as possible in the D part (the difficult
exercises). Any Ph. D. student interested in Abelian Group Theory
should solveall the exercises in our collection.
The first author acknowledges the Department ofMathematics and
Computer Science, Faculty of Science of Kuwait University for the
friendly environment created during the final stage of the manuscript
preparation.