UUttaahh SSttaattee UUnniivveerrssiittyy DDiiggiittaallCCoommmmoonnss@@UUSSUU All Graduate Plan B and other Reports Graduate Studies 5-1970 IInnfifinniittee AAbbeelliiaann GGrroouuppss Joaquin Pascual Follow this and additional works at: https://digitalcommons.usu.edu/gradreports RReeccoommmmeennddeedd CCiittaattiioonn Pascual, Joaquin, "Infinite Abelian Groups" (1970). All Graduate Plan B and other Reports. 1122. https://digitalcommons.usu.edu/gradreports/1122 This Report is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Plan B and other Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. 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 .