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The theory of groups Ian D. Macdonaid Thetheory ofgroups IAN D. MACDONALD The theory of groups OXFORD 'AT THE CLARENDON PRESS OxfordUniversityPrue,ElyHausa,LondonW.1 GLASGDW NBWYORKTORONTO MEI-300M?!Will-INCH“ CAPETOWNSALISIIUIYXBADAN NHBOBIDARisSAMAK IVSAKAADDISflm 80lCAMWI‘I‘AMAD“! KARACHI”MOREDACGA IVALALVIPURSINGAPOREBONDKONG TOKYO «3 0x501“)UNIVERSITYPRESS1968 rm“-runuaamx968 mammal:umoompmmuvmnunEmu}! mucannon»sung-rsorranrms’rEDITION u-an:umvnsnvmass,oxrolm myVIVIANmm,“ Hunt“tonu:umvznsxn 1970 Preface THEpresentworksprangfromacourseoflecturesgiventoan Australianpassclass,whenIfoundthatnoexistingelementary textreallymetmyneeds.Itbearsmanymarksofitsgenesisin thatfierycrucible.Ireferfirstofalltothepresentationofeven themostelementaryideasindetailandwiththesupportof many illustrations andexamples. Next, I have in mind the single-minded concentration on groups, to the exclusion of linkswithothermathematicsandofapplications;experience showsthatacourseofthiskindhastobeself-containedaswell assimple,andaglanceattheAppendixwillshowhowlittle materialisassumedknown:evenZorn’slemmaanditscognates arepassedoverinsilenceandneverused. Thethirdpointis thecarefulcollectionofproblemsforthereadertosolve,some invitinglyeasy,othersdifficultorworse.Naturallymypassstu- dentsreachedtheirdepthafteronlyafewchapters.Therestof the book followed on logically, and the subject-matter in particularselecteditself.Nodoubttheoutcomeisopentomany criticisms,butIhopethattheywillbetemperedbytheexperience ofgivingacoursesuchasmine. Itseemstomeevident,ifnotself-evident,thateventhehum— blestuniversitystudentshouldbeexposedtoalittleabstract algebra.Theproblemofwheretobreakintothesubjecthasan attractiveanswerinthetheoryofgroups.Butthisbelief,and theconsequentexistenceofthisbook,mustnotbeconstruedas adisparagementofanyotherbranchofmodernalgebra.Ifeel that,inparticular,thetheoryofgrouprepresentationsisofthe utmostimportancein developing thetheoryofgroups. This book contains as muchgroup theory as Iconsideritadvis— able tostudyinisolation; aftermasteringittheenterprising reader is urged to enrich his knowledge by broadening his algebraichorizons.Itmaybe’welltonoteherethatamodem vi Preface honourscoursemustcontainmuchotherabstractalgebraof importance. Afewcommentsonthemathematicsofthebookareaddres- sedtotheexpert.Anhonestattempthasbeenmadetoplace equalemphasisonfiniteandoninfinitegroups.Minimalaxioms forgroupshavebeentaken,bothtogivethenoviceexercise and becausetheyhave aplacein awork ofthistype. The main text doesnot contain the newproofs ofthestructure theoremforfinitelygeneratedabeliangroupsandofthethree Sylowtheorems:myfeelingis,onpedagogicgrounds,thattothe beginnertheyarehardlyeasierandprobablylessenlightening than the classicalproofs. Theproblemare usually common currencyamonggrouptheoristswhentheyarenotstandard. Finally,somewordsofgratitude.Anumberofmycolleagues and students have read the manuscript in part and made suggestions concerningit. Duringthesummerof1967—8the UniversityofQueensland made availablea JuniorResearch Assistantship,andMr.J.R.Ellwood,whooccupiedthisposi- tion,readthemanuscriptwithcareandsolvedalltheproblems. ThetypistswereMrs.MargaretF.BadenoftheUniversityof Newcastle,NewSouthWalesandMissPaulineC.Bakerofthe University of Queensland; their efiorts were considerable, skilful,andalwayscheerful,andMrs.Bodeninparticularpro- ducedmanyfinestencils.InthelaststageoftheworkIhave become much aware ofthe high professional standards and competence ofthe Clarendon Press. My expectations in this respectweremorethanfulfilled,andtheofficersandstaffhave been kind, helpful, and courteous without fail. To all these goodpeopleIoffermythanks. I.D.M. TheUniversityofQueensland Australia April1968 Contents —.. Someexamplesofgroups w. Basictheoremsandconcepts 16 . Complexesandsubgroups 34 w . Normalsubgroupsandfactorgroups 53 b l. Finitelygeneratedabeliangroups 82 l c. Normalstructure 102 é . Sylowtheorems 127 c w. Generatorsandrelations 149 . Nilpotentgroups 172 ._.O. Solublegroups 196 11. Surveyofexamples 213 REVISION PROBLEMS 230 APPENDIX 239 NOTE ONFURTHERREADING 249 INDEX 251