Infinite Abelian Groups

82 Pages·2017·9.94 MB·English

by  Joaquin Pascual

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All Graduate Plan B and other Reports Graduate Studies
5-1970
IInnfifinniittee AAbbeelliiaann GGrroouuppss
Joaquin Pascual
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RReeccoommmmeennddeedd CCiittaattiioonn
Pascual, Joaquin, "Infinite Abelian Groups" (1970). All Graduate Plan B and other Reports. 1122.
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INFINAIBTEEL IGARNO UPS
by
JoaquPians cual
A reposrutb mititnep da rtifaulli fllment
oft her equiremfeonrtt hsed egree
of
MASTEORF S CIENCE
in
Mat hmeat ci s
PlaBn
Approved:
UTASHT ATUEN IVERSITY
LogaUnt,a h
1970

NOTATION
Z Seto fi ntegers
Q Seto fr ationals
Zp:G rouopf i ntegmeord upl o
•, }a Setw hoseel emenatrsea ,. .. , a
n 1 n
.
. a, ] Subgroguepn erabtyea d, . .. , a
n 1 n
o(m)C yclgirco uopf o rdemr
00
o(p) :p -primcaormyp onoefnr ta tionmaoldsu lo one
tG Torsion suobfgG r oup
dG Maximdailv isisbulbeg rooufGp
G[ p ]: { x pE.x= G0 :}
nG:{ n:x x Ge}
1\
LJ\ Direscutm o ft heg roups( almoasltlc oordinaarteeO s)
k'=- K
ITJ\ Direpcrto duocftt heg rou�p s
kf=- K
ToA melia
TABLOEFC ONNTTSE
Page
INTRODUCTION 1
PRELIMIRNEASRUYL TS 2
INFINAIBTEEL IGANR OUPS 3
LITERACTIUTDRE E 75
VIT.A 76
1
INTRODUCTION
Whent het heoroyfg roupwsa sf irsitn troductehdea, t tention
waso nf initger oupsN.o wt,h ei nfiniatbee ligarno uphsa vec ome
inttoh eiorw n.T her esulotbst ainiendi nfiniatbee ligarno ups
arev eriyn terestainndpg e netratiinon tgh ebrr anchoefsM athe­
maticsF.o re xampleev,e rtyh eortehma ti ss tateidn t hisp aper
mayb eg eneralifzoerdm oduloevse rp rinicpali deadlo maiannsd
appliteodt hes tudoyf l ineatrr ansformations.
Thipsa peprr esentthsem osti mportraenstu litnsi nfinite
abeligarno upfso llowtihnege xpositgiiovne bny J .R otmainn h is
bookT,h eoroyf G roupsA:n I ntrotdiuocn.A lsos,o moef t hee xer­
cisegsi vebny J .R otmaanr ep resenitnet dh ispa perI.n o rdetro
facilitoautrse t udyt,w oc lassiicfatioonfsi nfiniatbee ligarno ups
areu sed.T hef irsrte ductehse s tudoyf a beligarno uptso t he
studoyf t orsigorno uptso,r sion-gfrroeuep asn da ne xtenspiorno blem.
Thes econcdl assificartediuocnte ost hes tudoyf d ivibsliea nd
reducgerdo upsF.o llowtihnigsi s a studoyf f reea beligarno uptsh at
are,i na certasienn sed,u atlo t hed ivisigbrloeu ptsh;e b asiasn d
fundmaentatlh eoreomfsf initegleyn eraatebde ligarno upasr ep roved.
Finalltyo,r sigorno upasn dt orsion-gfrroeuep osf r ank1 ares tudied.
Iti sa ssumtehda tt her eadeirs f amiliwairt he lementgarroyu p
theorayn df initaeb eligarno upsZ.o rn'lse mmias a ppliseedv eral
timeass w elals s omree sulotfsv ectosrp aces.
2
PRELIMIRNEASRULYT S
Thef ollowriensgu lwtisl ble u seidn t hes uppoorftt his
paper, arbeun ott d irectal pya rotf i t.
1.IfK andS areg roupasn,e xtensoifoK n b yS isa group
G sucthh at
a.G contaKianssa normaslu bgroup.
b.G/K"' S.
2.Every finite abGe ilsia a dni rgercsotuu mop f p -primary
group.
3.Everfyi niatbee lian Gg irsoa u dpi recsutm o fp rimary
cyclgirco ups.
4. IfG = �, thHen
i
i=l
mG= I mH.
l
i=l
wherme i sa positiivnet eger.
n
5.IfG = I H.,t hen
l
i=l
n
G[p] = I (H.[ p])
l
i=l
6.Evervye ctsopra chea sa basis.
7.Twob asefso ra vectsopra cVe h avteh es amneu mboefr
elemtesn.
3
INFINAIBTEEL IAN GROUPS
Allg roupusn decro nsideraartei ona belian wraintdt eanr e
additiveTlhyet. r ivigarlo uips t heo neh avionnge e lemeanntd
isd enotbeyd0 .
Definition
Int hef ollowdiinagg racma,p ital ldeetontteegr rso uapnsd
thea rrodwesn otheo momorphisms.
a
(3 I I
I
v
A'---- -- �B'
a'
Wes ayt hatth ed iagrcaomm mutiefS sa = a'S.'
Thef ollowiisno gn ee xampolfea commutdiianggr am
a
z
6
I
I ,1 1:
(3 i
CT (24) --- ]> 2
36
a'
wherze, z, andz aret heg roumpos du6l,o1 2a nd3 6r espectively
6 12 36
and0 (24i)s a cyclgirco uopf o rde2r4 .
CT
n ----i> 2n
4
s
m--�3
m
S' z�o(24)
6
3n
n ----a- -o-
wherei sta h eg eneraotfo0 r( 24).
3n
ConsidneorwS an() = S(na) S=( n2) =6n.a 'S(n') = a'( Sn') = a'( a) =
6n. Thne thea bovdei agrcaomm mutes.
Definition
A tringauladri agroafmt hef ollogw tiynpies a specitaylp oef
commnugt diiagram
p B
wherie i sa ni dnetithyo momorpnhdi scmo mamutgeis= fi;wf e a lso
sayt hagt extndesf .