STRUCTURE THEOREMS FOR INFINITE ABELIAN GROUPS A Thesis Presented to the Faculty of ...

56 Pages·2012·0.42 MB·English

STRUCTURETHEOREMSFOR
INFINITEABELIANGROUPS
AThesis
Presentedtothe
Facultyof
SanDiegoStateUniversity
InPartialFulfillment
oftheRequirementsfortheDegree
MasterofArts
in
Mathematics
by
JeremyGoodman
November,2012

iii
DEDICATION
DedicatedtoMichelle.
iv
Icallitpantbecauseit’sonepant.
–Cadence
v
ABSTRACT OF THE THESIS
StructureTheoremsfor
InfiniteAbelianGroups
by
JeremyGoodman
MasterofArtsinMathematics
SanDiegoStateUniversity,2012
Abeliangroupscanbeexaminedaccordingtotheirstructure. Oneofthestructureswe
willinvestigateisthatoffinitelygeneratedabeliangroups. Inparticular,thefundamental
theoremforfiniteabeliangroupsstatestheycanbeexpressedasadirectsumofprimary
cyclicgroups. Thesedecompositionsarenotnecessarilyunique,howevertheywillhavethe
sameinvariantfactors. Similarly,thereisafundamentaltheoremforfinitelygeneratedabelian
groupsstatesthatthesegroupscanbedecomposedintoadirectsumoffinitelymanycyclic
subgroups. However,thesesubgroupscanbefinite(cyclicp-groups)orinfinite(copiesof
(Z,+)). Justaswithfinitegroups,uniquenessisnotachieved,butdifferentdecompositions
willbeisomorphic. Thismeansthenumberofinfinitecyclicsummandswillbethesame,as
wellastheordersoftheprimarycyclicsummands,whichdefinetheinvariantsofthefinitely
generatedabeliangroup. Thepropertyofbeingfinitelygeneratedisclosedundersums,
subgroups,andquotients.
Twoimportantspecialclassesofinfiniteabeliangroupsaretorsiongroupsand
torsion-freegroups. Everytorsiongroupsplitsintoadirectsumofprimarygroups,andthis
decompositionisunique. Everyquotientgroupofanabeliangroupbyitstorsionsubgroupis
torsion-freegroup. So,everyabeliangroupisanextensionofatorsion-freegroupbyatorsion
group. Thisextensiondoesnotalwayssplit,buttherearecasesinwhichitdoes. Forexample,
thetorsionsubgroupofafinitelygeneratedabeliangroupisadirectsummand.
Anotherexamplewheretheextensionsplitsiswithdivisibleabeliangroups. Divisible
abeliangroupscanbedecomposedintoadirectsumofgroupsisomorphictotherationalsand
p∞ groupsforvariousprimesp. Divisibilityisclosedundersubgroups,quotients,direct
productsandsums.
vi
TABLE OF CONTENTS
PAGE
ABSTRACT .................................................................................... v
ACKNOWLEDGEMENTS.................................................................... vii
CHAPTER
1 GROUPS ............................................................................... 1
1.1 DefinitionsandNotation........................................................ 1
1.2 CyclicGroupsandDirectProducts ............................................ 5
1.3 IsomorphismTheorems......................................................... 9
1.4 CenterandCentralizer .......................................................... 14
1.5 ConjugatesandConjugacyClasses ............................................ 16
1.6 Cauchy’sTheorem .............................................................. 18
2 FINITEABELIANGROUPS ......................................................... 19
2.1 DefinitionsandNotation........................................................ 19
2.2 StructureTheoremsforFiniteAbelianGroups................................ 24
3 FINITELYGENERATEDABELIANGROUPS..................................... 31
3.1 TorsionandTorsion-FreeGroups .............................................. 31
3.2 FinitelyGeneratedAbelianGroups ............................................ 36
4 DIVISIBLEABELIANGROUPS .................................................... 41
4.1 DivisibleAbelianGroups....................................................... 41
BIBLIOGRAPHY.............................................................................. 49
vii
ACKNOWLEDGEMENTS
IwouldliketothankDr. Groneforhistime,energy,andcontinuedsupportoverthe
years. IwouldalsoliketothankDr. O’SullivanandDr. Lewisfortheirsupportaswell
throughoutanunconventionalprocess.
1
CHAPTER 1
GROUPS
Abeliangroupscanbeexaminedinanumberofways. Themethodweuseisto
determineunderwhatconditionsanabeliangroupsplitsbasedontheorderofeachelement
beingfiniteorinfinite. Wethenstudyeachsummandseparatelytoseeifitcanbedecomposed
further. Wewilllookatthreecategoriesofabeliangroups. Wefirstneedtounderstandthe
structureoffiniteabeliangroupsbeforeprogressingtoinfinitegroupsandsplittingbyorder.
Thetwocategoriesofinfiniteabeliangroupsweanalyzearefinitelygeneratedabeliangroups
anddivisiblegroups. Theultimategoalistobreakdownfinitelygeneratedgroupsintofinite
andinfinitecyclicgroups,anddivisiblegroupsintocopiesoftherationalsandp∞ groups.
Themaingoalofthischapteristoreviewsomefundamentalsingrouptheory. This
willincludebasicdefinitions,structuresofsubgroupsanddirectproducts,andisomorphisms,
allofwhichwillbeintegralinanalyzingbothfiniteandinfiniteabeliangroups.
1.1 DEFINITIONS AND NOTATION
Inthissectionwereviewsomebasicdefinitionsingrouptheory. Weintroducegroups,
subgroups,andnormality. Weusemultiplicativenotationtorepresentthebinaryoperationon
thegroup. Wewillalsoshowsomeexamplestoillustratetheseterms.
Definition1.1 AbinaryoperationonasetGisafunction∗ : G×G → G. Wedenotea∗b
tomean∗(a,b).
Definition1.2 LetGbeasetand∗abinaryoperation. Agroupisanorderedpair(G,∗)
thatsatisfiesthefollowingaxioms:
(i) ∗isassociative,i.e. (a∗b)∗c = a∗(b∗c)foralla,b,c ∈ G.
(ii) thereexistsanidentityelemente ∈ Gsuchthate∗a = a∗e = a,foralla ∈ G.
(iii) foreacha ∈ G,thereexistsaninverseofa,a−1 ∈ Gsuchthata∗a−1 = a−1 ∗a = e
Ingeneralwerefertothebinaryoperationofagroupasmultiplication,unless
otherwisestated. Also,tosimplifynotation,insteadofwritinga∗bforelementsinagroup,
wejustwriteabtodenotethemultiplication. Additionally,wewriteE tomean{e}.
Example1.1 Zisagroupunderusualintegeraddition,where0istheidentityelementand
forn ∈ Z,theinverseis−n. Here,weusea+b,ratherthana∗borab.
2
Q,R,andCarealsogroupsunderaddition. Theidentityandinversesaredefinedthe
sameasinZ.
Example1.2 Q\{0}isagroupunderusualmultiplication.
Theidentityforthisgroupis1,andforalla ∈ Q\{0},a−1 is 1. Futhermore,R\{0}
a
andC\{0}arealsogroupsundermultiplication,withthesameidentityanddefinitionof
inverses. However,Z\{0}isnotagroupundermultiplication,althoughitisassociativeand
doeshaveanidentity,sinceitlackstheexistenceofinverses. Forexample,2 ∈ Z\{0},but
1 ∈/ Z\{0}.
2
Example1.3 LetG = {−1,1}. ThenGisagroupunderusualmultiplication.
AssociativityandtheidentityareinheritedfromQ\{0}. EachelementofGisitsown
inverse.
Example1.4 LetΩbeanonemptyset,andS bethesetofallpermutationsofΩ. S isa
Ω Ω
groupundercompositionoffunctions.
Weshouldfirstnotethat◦isabinaryoperationonS . Ifσ andτ areboth
Ω
permutationsofΩ,thensoisσ ◦τ. Compositionoffunctionsisassociativeingeneral. The
identityelement,e,isthepermutationdefinedbye(ω) = ω forallω ∈ Ω. Andforevery
σ ∈ S ,thereisaninversepermutationsuchthatσ ◦σ−1 = σ−1 ◦σ = e.
Ω
Afundamentalresultinthetheoryofgroupsinvolvespermutations. Specifically,a
theoremofCayleyStatesthateverygroupGisisomorphictoasubgroupofS . Wewilllater
G
definewhatitmeanstobeisomorphic,butthemappingcanbedescribedby
g (cid:55)→ ϕ ,
g
whereϕ (h) = gh, allh ∈ G
g
Itisstraightforwardtoshowthatϕ givesapermutationG. Furthermore,ϕ = ϕ ϕ and
g g1g2 g1 g2
(ϕ )−1 = ϕ .
g g−1
Themappingg (cid:55)→ ϕ playsanimportantroleinthetheoryofgrouprepresentations,
g
andisknownastheleftregularrepresentationofG.
IfΩ = {1,2,...,n},thenwecallS thesymmetricgroupofdegreen. Toestablish
n
theorderofS ,wemustlookatthepossiblepermutations. Letσ ∈ S . Thentherearen
n n
possibleplacesσ cansend1. Then,thenumberofplacesforσ(2)isn−1astheonlyplaceit
cannotbeisσ(1). Clearly,σ(3)hasn−2possibilities,andsoonuptoσ(n)havingonlyone
placeitcanbe. So,thereareexactlyn!permutationsofΩ. Hence,theorderofthisgroupisn!.
3
(cid:40) (cid:32) (cid:33) (cid:41)
a a
Example1.5 LetG = A = 1 3 : a ,a ,a ,a ∈ R,det(A) (cid:54)= 0 . ThenGisa
1 2 3 4
a a
2 4
groupunderusualmatrixmultiplicaiton.
ItcaneasilybeverifiedthatGisclosedundermatrixmultiplicationandthatthe
(cid:32) (cid:33)
1 0
associativelawholdsaswell. TheidentityofGisI = . I isclearlyinGas
0 1
1·1−0·0 = 1 (cid:54)= 0,andAI = IA = AforallA ∈ G.
InorderforGtobeagroup,foreveryA ∈ G,theremustexistanA−1 ∈ Gsuchthat
AA−1 = A−1A = I. Thematrixdefinedby
(cid:32) (cid:33)
a4 −a3
A−1 = a1a4−a2a3 a1a4−a2a3
−a2 a1
a1a4−a2a3 a1a4−a2a3
indeeddoessatisfythenecessaryproperties. Ofcoursethereisnothingspecialaboutthesize
ofthematrixandorthefieldofrealnumbers. Wecandefineagroupofmatricesofsizen
overageneralfieldF. Thisiscalledthegenerallineargroupofdegreen,writtenGL (F).
n
Example1.6 Let
GL (F) = {A : Aisann-by-nmatrixwithentriesfromafieldF and det(A) (cid:54)= 0}. Then
n
GL (F)isagroupundermatrixmultiplication.
n
Definition1.3 AnonemptysubsetH ofagroup,G,isasubgroupifH formsagroupunder
thebinaryoperationinG. H isapropersubgroupofGiffirstitisasubgroup,andsecond,
isapropersubsetofG.
IfH isasubgroupofG,wewillwriteH ≤ G. IfH isapropersubgroupofG,we
willwriteH < G.
Lemma1.1 LetGbeagroup. ThenH,anonemptysubsetofG,isasubgroupofGifand
onlyif:
(i) H isclosedundermultiplication,i.e. foralla,b ∈ H,ab ∈ H,
and
(ii) H isclosedunderinverses,i.e. foralla ∈ H,a−1 ∈ H.
Proof: IfH isasubgroupofG,thenbydefinitionH formsagroupbyitself,anditisclear
thatitisclosedundermultiplicationandinverses. Heretheterm”multiplication”means
additionofintegers. Forthereversedirection,theassociativityandcommutativityare
inheritedfromG. IfH isclosedunderinverses,thenifa ∈ Gwehavea−1 ∈ G. SinceH is
closedundermultiplication,a−1a = aa−1 = e ∈ G,sotheidentityofH isthesameidentity
ofG. Therefore,H isasubgroupofG (cid:3)