Infinite Groups: A Roadmap to Selected Classical Areas

411 Pages·2023·19.676 MB·English

by  Martyn R. Dixon, Igor Ya. Subbotin, Leonid A. Kurdachenko

Infinite Groups
In recent times, group theory has found wider applications in various fields of algebra and math-
ematics in general. But in order to apply this or that result, you need to know about it, and such
results are often diffuse and difficult to locate, necessitating that readers construct an extended
search through multiple monographs, articles, and papers. Such readers must wade through the
morass of concepts and auxiliary statements that are needed to understand the desired results,
while it is initially unclear which of them are really needed and which ones can be dispensed
with. A further difficulty that one may encounter might be concerned with the form or language
in which a given result is presented. For example, if someone knows the basics of group theory,
but does not know the theory of representations, and a group theoretical result is formulated in
the language of representation theory, then that person is faced with the problem of translating
this result into the language with which they are familiar, etc.
Infinite Groups: A Roadmap to Selected Classical Areas seeks to overcome this challenge.
The book covers a broad swath of the theory of infinite groups, without giving proofs, but with
all the concepts and auxiliary results necessary for understanding such results. In other words,
this book is an extended directory, or a guide, to some of the more established areas of infinite
groups.
Features
• An excellent resource for a subject formerly lacking an accessible and in-depth reference
• Suitable for graduate students, PhD students, and researchers working in group theory
• Introduces the reader to the most important methods, ideas, approaches, and constructions
in infinite group theory.
Taylor & Francis
Taylor & Francis Group
http://taylorandfrancis.com
Infinite Groups
A Roadmap to Selected Classical Areas

Martyn R. Dixon
University of Alabama, United States of America
Leonid A. Kurdachenko
Oles Honchar Dnipro National
University, Ukraine
Igor Ya. Subbotin
National University, USA
First edition published 2023
by CRC Press
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CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin
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ISBN: 978-0-367-70262-5 (hbk)
ISBN: 978-0-367-70697-5 (pbk)
ISBN: 978-1-003-14757-2 (ebk)
DOI: 10.1201/9781003147572
Typeset in Latin Modern
by KnowledgeWorks Global Ltd.
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Contents
Preface xi
Authors xvii
Chapter 1(cid:4) Important Subgroups 1
1.1 SOMEIMPORTANTSERIESINGROUPSANDSUBGROUPS
DEFINEDBYTHESESERIES 4
1.2 CLASSESOFGROUPSDEFINEDBYSERIESOFSUBGROUPS 12
1.3 RADICABLEGROUPS 18
1.4 SOMETHINGFROMTHETHEORYOFMODULES 21
1.5 THE0-RANKANDp-RANKOFABELIANGROUPS 22
1.6 THEFRATTINISUBGROUPOFAGROUP 25
1.7 LINEARGROUPS 28
1.8 RESIDUALLYX-GROUPS 33
References for Chapter 1 41
Chapter 2(cid:4) Finitely Generated Groups 49
2.1 THEGENERALIZEDBURNSIDEPROBLEM 51
2.2 THEBURNSIDEPROBLEMFORGROUPSOFFINITEEXPONENT 52
2.3 THERESTRICTEDBURNSIDEPROBLEM 54
2.4 GROWTHFUNCTIONSONFINITELYGENERATEDGROUPS 55
2.5 FINITELYPRESENTEDGROUPS 58
2.6 GROUPSWITHTHEMAXIMALCONDITIONFORALLSUBGROUPS 61
References for Chapter 2 72
Chapter 3(cid:4) Finiteness Conditions 83
3.1 THEMINIMALCONDITIONONCERTAINSYSTEMSOF
SUBGROUPS 83
vii
viii (cid:4) Contents
3.2 THEMINIMALCONDITIONONNORMALSUBGROUPS 90
3.3 ARTINIANANDRELATEDMODULESOVERSOMEGROUPRINGS 93
3.4 MINIMAXGROUPS 102
3.5 THEWEAKMINIMALCONDITION 107
3.6 THEWEAKMAXIMALCONDITION 114
References for Chapter 3 117
Chapter 4(cid:4) Ranks of Groups 127
4.1 FINITESPECIALRANKANDFINITESECTIONp-RANK 127
4.2 FINITE0-RANK 130
4.3 THECONNECTIONSBETWEENTHEVARIOUSRANK
CONDITIONSI 133
4.4 FINITESECTIONRANK 134
4.5 BOUNDEDSECTIONRANK 138
4.6 THECONNECTIONSBETWEENTHEVARIOUSRANK
CONDITIONSII 139
4.7 FINITELYGENERATEDGROUPS 143
4.8 SYSTEMSOFSUBGROUPSSATISFYINGRANKCONDITIONS 146
4.9 SOMERESIDUALSYSTEMS 149
References for Chapter 4 153
Chapter 5(cid:4) Conjugacy Classes 159
5.1 AROUND “SCHUR’S THEOREM”, CENTRAL-BY-FINITE GROUPS
ANDRELATEDTOPICS 160
5.2 BOUNDEDCONJUGACYCLASSES,FINITE-BY-ABELIANGROUPS
ANDRELATEDCLASSES 172
5.3 GROUPSWITHFINITECLASSESOFCONJUGATEELEMENTS 177
5.4 SOMECONCLUDINGREMARKS 192
References for Chapter 5 193
Chapter 6(cid:4) Generalized Normal Subgroups and their Opposites 203
6.1 GROUPSWHOSESUBGROUPSARENORMAL,PERMUTABLEOR
SUBNORMAL 204
6.2 GROUPSHAVINGALARGEFAMILYOFNORMALSUBGROUPS 211
Contents (cid:4) ix
6.3 GROUPSHAVINGALARGEFAMILYOFSUBNORMAL
SUBGROUPS 218
6.4 PAIRSOFOPPOSITESUBGROUPS 225
6.5 TRANSITIVELYNORMALSUBGROUPS 234
6.6 THENORMOFAGROUP,THEWIELANDTSUBGROUPAND
RELATEDTOPICS 243
6.7 THENORMOFAGROUPANDTHEQUASICENTRALIZER
CONDITION 249
References for Chapter 6 254
Chapter 7(cid:4) Locally Finite Groups 269
7.1 PRELIMINARIES 269
7.2 LARGELOCALLYFINITEGROUPS 272
7.3 SIMPLELOCALLYFINITEGROUPS 275
7.4 EXISTENTIALLYCLOSEDGROUPS 282
7.5 CENTRALIZERSINLOCALLYFINITEGROUPS 284
7.6 SYLOWTHEORYINLOCALLYFINITEGROUPS 288
7.7 CONJUGACYOFSYLOWSUBGROUPS 290
7.8 UNCONVENTIONALSYLOWTHEORIES 294
7.9 SATURATEDFORMATIONSANDFITTINGCLASSES 297
7.10 BARELYTRANSITIVEGROUPS 301
References for Chapter 7 303
Bibliography 315
Author Index 377
Symbol Index 381
Subject Index 385