Classifications of Infinite Abelian Groups (De Gruyter Studies in Mathematics)

342 Pages·2019·1.884 MB·English

by  Carol Jacoby

Carol Jacoby and Peter Loth
Abelian Groups
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Structures and Classifications
MathematicsSubjectClassification2010
Primary:20-2,20Kxx,03C52,13C05;Secondary:03C60,03E10,22B05,22D35
Authors
Dr.CarolJacoby
JacobyConsulting
LongBeach,CA
USA
[e-mail]
Prof.Dr.PeterLoth
DepartmentofMathematics
SacredHeartUniversity
5151ParkAvenue
Fairfield,CT06825
USA
[e-mail]
ISBN978-3-11-043211-4
e-ISBN(PDF)978-3-11-042768-4
e-ISBN(EPUB)978-3-11-042786-8
ISSN0179-0986
LibraryofCongressControlNumber:2019938386
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
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|
ForourspousesJerryandAlice
InmemoriamRüdigerGöbel

Preface
How do you get your arms around an infinite abelian group? Such groups are all
aroundus—integers,rationals,reals,andcomplexnumbers.Yettheseexamplesdo
notevenhintattherichnessandcomplexitiesofabeliangroups,aswewillsee.To
helpingettingthroughthecomplexitywewanttofocusinonsomesetofcharacter-
isticsthatcompletelydeterminesthegroup.Specifically,weseeksomeinvariantsof
agroupsuchthatiftwogroupssharethesameinvariants,thentheyareofnecessity
isomorphic.
Asitturnsout,thisproblemcannotbesolvedingeneral.Thereisatrade-offbe-
tweenthesizeoftheclassofgroupsclassifiedandthecomplexityoftheinvariants.
Thismeansthatakeypartoftheclassificationtaskisfindingclassesofgroupsthat
admittoaninsightfulclassification.Thus,theclassificationproblemhasthreeparts:
(1)Identifyaclassofgroupstobeclassified.Thebroadertheclassthebetter.Much
oftheworkinthisareainvolvesexpandingtheclass.Forexample,Ulmwasableto
classifycountabletorsiongroups,andlaterworkexpandedtheclassificationtoother
torsiongroups,andthentocertainmixedgroups(thatis,thosethatareneithertorsion
nortorsion-free)withadditionalinvariants.Theidealclasswillbeclosedunderkey
operations to aid studying it. In particular, the class should be closed under direct
sumsandsummands,andideallyGshouldbeintheclassifandonlyifbothpαGand
G/pαGareintheclass.
(2) Identify an invariant, usually a cardinal, that can be determined for each
group.Itissomewhatmisleadingtosay“aninvariant”sinceitmayactuallyconsist
of infinitely many values. Prove that it is indeed an invariant by proving that it is
unchangedunderisomorphism.
(3)Provethattheinvariantclassifiesthegroupsintheclass.Specifically,prove
thatiftwogroupsintheclasshavethesameinvariants,thentheyareisomorphic.In
somecases,wecanexpandtheclassbydemandingsomethinglessthanfullisomor-
phism.
Thisexplorationleadstothediscoveryofinterestingclassesofgroups,andhence
tothestructureproblem.Thegoalistodescribethestructureorotherdefiningchar-
acteristicsofaclassofgroups.Forexample,wewillseethatgroupsincertainclasses
maybewrittenandthoughtofasdirectsumsofcertaincyclicgroups.Oftenthisstruc-
turesuggestsinvariantstouseforclassification.Structureandclassificationareof-
tencombinedintouniquenesstheoremsthatpresenttheclassastheuniqueonethat
containscertaingroups,isclosedundercertainoperationsandclassifiedbycertain
invariants.
ItcanbearguedthatGaussgaveusthefirstclassificationresultin1801,thefi-
niteversionoftheprimarydecompositiontheorem.Thiswasevenbeforegroupsand
abelian groups were defined by Kronecker in 1870. The first classification result of
infinitegroupswasduetoUlmin1933.Hedevelopedcardinalinvariantsthatcom-
https://doi.org/10.1515/9783110427684-201
VIII | Preface
pletelyclassifiedcountabletorsiongroups.Soonafter,Baer(1937)developedinvari-
antscalledtypesthatclassifytorsion-freegroupsofrank1.Thiswasthefirstwork
thatsystematicallyaddressedgroupsthatwerenotnecessarilycountable.Interestin
abeliangroupsreignitedinthemid-20thcenturywiththepublicationofbooksonthe
topic[78,24]byKaplansky(1954)andFuchs(1958),thelattercontaining86openprob-
lems.HillextendedUlm’sclassificationtoalargerclassoftorsiongroups,thetotally
projectivegroups.Thesetorsionandtorsion-freeresultsseemedtobeunrelateduntil
Warfielddevelopedaclassificationtheoryofaclassofgroupsthatincludesboth,but
alsosomegroupsthatdonotsplitintoadirectsumoftorsionandtorsion-freesub-
groups.Anapparentwaytolookatthesemixedgroupsisasextensionsoftheirtor-
sionsubgroupbyatorsion-freegroup.Warfieldflippedthisaroundandviewedthese
mixedgroupsintermsofatorsion-freebasis,characterizedbytheWarfieldinvariants,
withthequotientoverthisbasistorsionandtotallyprojective,andthuscharacterized
bytheUlminvariants.
Weextendthisworkintwoways.Oneistoweakentherequirementforisomor-
phism.BarwiseandEklofextendedUlm’stheoremtoalltorsiongroupsuptopartial
isomorphism.Sincepartialisomorphismimpliesisomorphismoncountablegroups,
thisisatruegeneralization.InChapter5,wedefineaclassofgroupsthatincludesthe
Warfieldgroupsandclassifyituptopartialisomorphism.Partialisomorphismhasan
interestingmodel-theoreticcounterpartthatweexplore.
Theotherextensionlooksattopologicalgroups.InChapters6and7,weusePon-
trjagindualitytodeterminethedualcounterpartsofvariousconstructsintopological
groups.Everygroupresulthasacorrespondingresultincompactabeliangroups.In
particular,thisgivesusclassificationsforvariousclassesofcompactabeliangroups.
Theclassesofgroupsallhavearangeofcharacterizations.Weexploreeachclass
intermsofpresentations,homology,increasingsequencesofsubgroups,decomposi-
tionbases,andmanyothers.Manyofthechaptersculminateinalistofvariedchar-
acterizationsthatareprovedequivalent.
Weassumethereaderisfamiliarwiththedefinitionofgroup,andkeyconstructs,
suchassubgroups,factorgroups,homomorphisms,andisomorphisms.Whenweuse
the word “group”, we will mean additive abelian group. At times, we will consider
ℤ
groupsasmodulesovertheintegersorover ,aringthatwillbeusefulforlookingat
p
groupsoneprimeatatime.Necessarybackgroundinthekeycharacteristicsofinfinite
abeliangroupsisprovidedinChapter1.
Acknowledgments
WewouldliketoexpressourgratitudetothememoryofthelateProfessorRüdiger
GöbeloftheUniversityofDuisburg–Essen.HegreatlyimpactedthedirectionofPeter
Loth’sresearchandinitiatedaresearchprojectthatledtooursubsequentcollabora-
tion.CarolJacobywouldliketothankhimforhisgeneroushelpandencouragementin
incorporatingandextendingherpreviousresearch.ShegratefullyacknowledgesPro-
fessorEmeritusPaulEklofoftheUniversityofCalifornia,Irvine,forrecognizingthe
synergypossiblebetweenthisresearchandherpreviouswork,andprovidingsugges-
tionsandconnectionstomakeithappen.
This project was partially supported by a University Research/Creativity Grant
(URCG)fromSacredHeartUniversity.WewishtothankWalterdeGruyterVerlagfor
thepublicationofthisbook.
https://doi.org/10.1515/9783110427684-202