ToMary,Je,Michael,RebeccaandCliordGeorge M. BergmanAn invitation to General Algebraand Universal ConstructionsApril2,2015SpringerContents1 Aboutthecourse,andthesenotes . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Aimsandprerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Aquestionaday. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Thenameofthegame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Otherreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Numeration;history;advice;webaccess;corrections . . . . . . . 61.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7PartI.Motivationandexamples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Makingsomethingsprecise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Whatisagroup? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Indexedsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Arity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Group-theoreticterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Termsinotherfamiliesofoperations . . . . . . . . . . . . . . . . . . . . . 193 Freegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Thelogiciansapproach:constructionfromterms . . . . . . . . . . 253.3 Freegroupsassubgroupsofbigenoughdirectproducts . . . . 293.4 Theclassicalconstruction:groupsofwords . . . . . . . . . . . . . . . 344 ACookstour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Imposingrelationsongroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Presentationsofgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Abeliangroupsandabelianizations . . . . . . . . . . . . . . . . . . . . . . 524.5 TheBurnsideproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Productsandcoproductsofgroups . . . . . . . . . . . . . . . . . . . . . . 584.7 Productsandcoproductsofabeliangroups . . . . . . . . . . . . . . . 654.8 Rightandleftuniversalproperties . . . . . . . . . . . . . . . . . . . . . . . 674.9 Tensorproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.10 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.11 Groupstomonoidsandback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.12 Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83viiviii Contents4.13 Coproductsandtensorproductsofrings . . . . . . . . . . . . . . . . . 914.14 BooleanalgebrasandBooleanrings . . . . . . . . . . . . . . . . . . . . . . 954.15 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.16 Somealgebraicstructureswehavenotlookedat . . . . . . . . . . . 994.17 Stone-Cechcompactication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.18 Universalcoveringspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107PartII.Basictoolsandconcepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 Orderedsets,induction,andtheAxiomofChoice. . . . . . . . 1115.1 Partiallyorderedsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Preorders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 Induction,recursion,andchainconditions. . . . . . . . . . . . . . . . 1245.4 Theaxiomsofsettheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.5 Well-orderedsetsandordinals . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.6 ZornsLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.7 Thoughtsonsettheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586 Lattices,closureoperators,andGaloisconnections . . . . . . . 1616.1 Semilatticesandlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.2 Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.3 Closureoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.4 Apatternofthrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.5 Galoisconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917 Categoriesandfunctors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.1 Whatisacategory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.3 Othernotationsandviewpoints . . . . . . . . . . . . . . . . . . . . . . . . . 2147.4 Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.5 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.6 Contravariance,andfunctorsofseveralvariables . . . . . . . . . . 2327.7 Propertiesofmorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2387.8 Specialobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2477.9 Morphismsoffunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2607.10 Propertiesoffunctorcategories . . . . . . . . . . . . . . . . . . . . . . . . . 2707.11 Enrichedcategories(asketch) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2738 Universalconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.1 Initialandterminalobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.2 Representablefunctors,andYonedasLemma. . . . . . . . . . . . . 2788.3 Adjointfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2878.4 Thep-adicnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988.5 Directandinverselimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3058.6 Limitsandcolimits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3148.7 Whatrespectswhat? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Contents ix8.8 Functorsrespectinglimitsandcolimits . . . . . . . . . . . . . . . . . . . 3258.9 Interactionbetweenlimitsandcolimits . . . . . . . . . . . . . . . . . . . 3338.10 Someexistencetheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.11 Morphismsinvolvingadjunctions. . . . . . . . . . . . . . . . . . . . . . . . 3488.12 Contravariantadjunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3549 Varietiesofalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3589.1 Thecategory-Alg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3589.2 Generatingalgebrasfrombelow. . . . . . . . . . . . . . . . . . . . . . . . . 3669.3 Termsandleftuniversalconstructions . . . . . . . . . . . . . . . . . . . 3699.4 Identitiesandvarieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3779.5 Derivedoperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3889.6 Characterizingvarietiesandequationaltheories . . . . . . . . . . . 3929.7 Liealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4029.8 Someinstructivetrivialities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4109.9 Clonesandclonaltheories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4129.10 StructureandSemantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425PartIII.Moreonadjunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43110 Algebras,coalgebras,andadjunctions . . . . . . . . . . . . . . . . . . . . 43210.1 Anexample: SL(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43210.2 Algebraobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43610.3 Coalgebraobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44110.4 Freydscriterionforexistenceofadjoints . . . . . . . . . . . . . . . . . 44510.5 Somecorollariesandexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 44810.6 Endofunctorsof Monoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45410.7 Functorstoandfromsomerelatedcategories . . . . . . . . . . . . . 46510.8 Abeliangroupsandmodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.9 Leftadjointfunctorsonmodules . . . . . . . . . . . . . . . . . . . . . . . . 47410.10 Generalresults,mostlynegative . . . . . . . . . . . . . . . . . . . . . . . . . 48010.11 Afewideasandtechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48410.12 Contravariantrepresentablefunctors . . . . . . . . . . . . . . . . . . . . . 48910.13 Moreoncommutingoperations . . . . . . . . . . . . . . . . . . . . . . . . . 49610.14 Furtherreading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503WordandPhraseIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505SymbolIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525ListofExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539end . . . . . . . . . . . . . . . . . . . . . . . . 545Chapter1Aboutthecourse,andthesenotes1.1. AimsandprerequisitesThiscoursewill developsomeconceptsandresultswhichoccurrepeatedlythroughout the various areas of algebra, andsometimes inother elds ofmathematics, and which can provide valuable tools and perspectives to thoseworking in these elds. There will be a strong emphasis on motivation throughexamples,andoninstructiveexercises.I will assume only an elementary background in algebra, corresponding toan honors undergraduate algebra course or one semester of graduate algebra,plus amoderate level of mathematical sophistication. Astudent whohasseen the concept of free group introduced, but isnt sure he or she thoroughlyunderstood it would be in a ne position to begin. On the other hand, anyoneconversant with fewer than three ways of proving the existence of free groupshassomethingtolearnfromChapters2-3.Asageneral rule, wewill payattentiontopettydetailswhentheyrstcomeup, buttakethemforgrantedlateron. Sostudentswhondthebe-ginningsectionsdevotedtoomuchtotriviashouldbepatient!In preparing this published version of my course notes, I have not removedremarks about homework, course procedures etc., addressed to students whotake the course from me at Berkeley, which take up the next three and a halfpages, since there are some nonstandard aspects to the way I run the course,whichI thoughtwouldbeof interesttoothers. Anyoneelseteachingfromthis text should, of course, let his or her students know which, if any, of theseinstructionsapplytothem.Inanycase,Ihopereaderselsewherendthesepagesmoreamusingthanannoying.12 1 Aboutthecourse,andthesenotes1.2. ApproachSince I tookmyrst graduate course, it has seemedtome that there issomethingwrongwithourmethodofteaching.Why,foranhouratatime,shouldaninstructor writenotes onablackboardandstudents copythemintotheirnotebooksoftentoobusywiththecopyingtopayattentiontothecontentwhenthisworkcouldbedonejustaswell byaphotocopyingmachine?Ifthisisallthathappensintheclassroom,whynotassignatextordistributeduplicatednotes,andrunmostcoursesasreadingcourses?Oneansweristhatthisisnotall thathappensinaclassroom. Studentsaskquestionsaboutconfusingpointsandtheinstructoranswersthem. So-lutions toexercises arediscussed. Sometimes aresult is developedbytheSocratic method through discussion with the class. Often an instructor givesmotivation, orexplainsanideainanintuitivefashionheorshewouldnotputintoawrittentext.Asforthislastpoint, Ithinkoneshouldnotbeembarrassedtoputmo-tivationandintuitive discussionintoatext, andI have includedagreatdeal of bothinthesenotes. Inparticular, Ioftenrstapproachgeneral re-sultsthroughveryparticularcases. Theotheritemsansweringquestions,discussingsolutionstoexercises,etc.whichseemtometocontainthees-sential humanvalueof classcontact, arewhatIwouldlikeclassroomtimetobespentoninthiscourse, whilethesenoteswill replacethemechanicalcopyingofnotesfromtheboard.Such a system is not assured of success. Some students may be in the habitoflearningmaterial throughtheprocessofcopyingit,andmaynotgetthesamebenetbyreadingit.Iadvisesuchstudentstoreadthesenoteswithapad of paper in their hands, and try to anticipate details, work out examples,summarizeessential points, etc., astheygo. Myapproachalsomeansthatstudents needtoreadeachdays material beforetheclass whenit will becovered,whichmanystudentsarenotaccustomedtodoing.1.3. AquestionadayTohelpthesystemdescribedaboveworkeectively,Irequireeverystudenttakingthiscoursetohandin,oneachdayofclass,onequestionconcerningthe reading for that day. I strongly encourage you to get your question to mebye-mail byatleastanhourbeforeclass. If youdo, Iwill trytoworktheanswerintowhatIsayinclassthatday.Ifnot,thenhanditinatthestartofclass, andIwill generallyansweritbye-mail ifIfeel Ididnotcoverthepointinclass.Thee-mail orsheetof paperwithyourquestionshouldbeginwithyourname, the point in these notes that your question refers to, and the classifyingwordurgent,important,unimportantorproforma.Therstthree1.4 Homework 3choices of classifyingwordshouldbeusedtoindicatehowimportant it istoyoutohavethequestionanswered; usethelastoneiftherewasnothinginthereadingthat youreallyfelt neededclarication. Inthat case, yourproformaquestionshouldbeonethatsomereadermightbepuzzledby;perhapssomethingthatpuzzledyouatrst, butthatyouthenresolved. Ifyougiveaproformaquestion,youmustgivetheansweralongwithit!Youmayaskmorethanonequestion; youmayask, inadditiontoyourquestiononthecurrentreading, questionsrelatingtoearlierreadings, andyouareencouragedtoaskquestionsinclassaswell. Butyoumustalwayssubmitinwritingatleastonequestionrelatedtothereadingassignmentfortheday.1.4. HomeworkThesenotescontainalargenumberofexercises.Iwouldlikeyoutohandinsolutions to an average of one or two problems of medium diculty per week,oracorrespondinglysmallernumberofharderproblems,oralargernumberof easier problems. Choose problems that interest you. But please, look at alltheexercises,andatleastthinkabouthowyouwouldapproachthem.Theyare interspersed through the text; you may prefer to think about some as youcometothem, andtocomebacktoothersafteryounishthesection. Wewill discuss many of them in class. I recommend spending at least one to veminutesthinkingabouteachexercise,unlessyouseeasolutionimmediately.Grades will be basedlargelyonhomework. The amount of homeworksuggested above, reasonably well done, will give an A. I will give partial creditforpartial results, aslongasyoushowyourealizethattheyarepartial. Iwouldalsowelcomeyour bringingtotheattentionof theclass interestingrelatedproblemsthatyouthinkof,orndinothersources.It shouldhardlyneedsayingthat asolutiontoahomeworkexerciseingeneral requires aproof. If aproblemasks youtondanobject withacertainproperty, itisnotsucienttogiveadescriptionandsay, Thisisthedesiredobject; youmustprovethatithastheproperty, unlessthisiscompletelyobvious. If aproblemasks whether acalculationcanbe donewithoutacertainaxiom, itisnotenoughtosay, No, theaxiomisusedinthecalculation; youmustprovethatnocalculationnotusingthataxiomcanleadtotheresultinquestion. If aproblemaskswhethersomethingistrueinall cases, andtheanswerisno, thentoestablishthisyoumust, ingeneral,giveacounterexample.Iamworriedthattheamountof handwaving(informal discussion)inthesenotesmayleadsomestudentstothinkhandwavingisanacceptablesubstitutefor proof. If youreadthesenotes attentively, youwill seethathandwavingdoesnotreplaceproofs. Iuseittoguideustoproofs, tocom-municatemyunderstandingofwhatisbehindsomeproofs,andattimesto4 1 Aboutthecourse,andthesenotesabbreviateaproofwhichissimilartoonewehavealreadyseen;butincasesof the last sort there is a tacit challenge to you, to think through whether youcanindeedllinthesteps.Homeworkismeanttodevelopanddemonstrateyourmasteryof thematerial andmethods, soitisnot aplaceforyoutofollowthismodelbychallengingtheinstructortollinsteps!Of course, thereis alimit totheamount of detail youcanandshouldshow. Mostnontrivial mathematical proofswouldbeunreadableifwetriedto give every substep of every step. So truly obvious steps can be skipped, andfamiliarmethodscanbeabbreviated.Butmorestudentserrinthedirectionofincompleteproofsthanofexcessivedetail.Ifyouhavedoubtswhethertoabbreviateastep, thinkout(perhapswiththehelpofascratch-pad)whatwouldbeinvolvedinamorecompleteargument. If youndthatastepismorecomplicatedthanyouhadthought,thenitshouldnotbeomitted!Butbearinmindthattoshowornottoshowamessystepmaynotbetheonlyalternativesbeonthelookoutforasimplerargument,thatwillavoidthemessiness.I will try to be informative in my comments on your homework. If you arestillindoubtastohowmuchdetailtosupply,cometomyoceanddiscussit. If possible, come with a specic proof in mind for which you have thoughtoutthedetails,butwanttoknowhowmuchshouldbewrittendown.There are occasional exceptions to the principle that every exercise requiresa proof. Sometimes I give problems containing instructions of a dierent sort,such as Write down precisely the denition of . . . , or State the analogousresult in the case . . . , or How would one motivate . . . ? Sometimes, once anobject with a given property has been found, the verication of that propertyistrulyobvious.However,ifdirectvericationofthepropertywouldinvolve32caseseachcomprisinga12-stepcalculation,youshould,ifatallpossible,ndsomeargumentthatsimpliesoruniesthesecalculations.Exercises frequently consist of several successive parts, and you may handin some parts without doing others (though when one part is used in another,youshouldifpossibledotheformerifyouaregoingtodothelatter). Thepartsofanexercisemayormaynotbeofsimilardicultyonepartmaybeaneasyverication,leadinguptoamoredicultpart;oranexerciseofmoderatedicultymayintroduceanopenquestion.(Openquestions,whenincluded,arealwaysnotedassuch.)Homeworkshouldbelegibleandwell-organized. If asolutionyouhaveguredoutiscomplicated, oryourconceptionof itisstill fuzzy, outlineitrst on scratch paper, and revise the outline until it is clean and elegant beforewriting up the version to hand in. And in homework as in other mathematicalwriting, when an argument is not completely straightforward, you should helpthe reader see where you are going, with comments like, We shall now prove. . . ,Letusrstconsiderthecasewhere. . . ,etc..Ifyouhandinaproofthatisincorrect,Iwillpointthisout,anditisuptoyouwhethertotrytondandhandinabetterproof. If, instead, Indtheproofpoorlypresented,Imayrequirethatyouredoit.1.6 Otherreading 5Ifyouwanttoseethesolutiontoanexercisethatwehaventgoneover,askinclass. Imaypostponeanswering, orjustgiveahint, if otherpeoplestill wanttoworkonit. Inthecaseof anexercisethatasksyoutosupplydetails for the proof of a result in the text, if you cannot see how to do it youshouldcertainlyasktoseeitdone.You may also ask for a hint on a problem. If possible, do so in class ratherthaninmyoce,sothateveryonehasthebenetofthehint.If twoor moreof yousolveaproblemtogether andfeel youhavecon-tributed approximately equal amounts to the solution, you may hand it in asjointwork.If you turn in a homework solution which is inspired by material you haveseeninanothertextorcourse, indicatethis, sothatcreditcanbeadjustedtomatchyourcontribution.1.5. ThenameofthegameThe general theoryof algebraic structures has longbeencalledUniversalAlgebra, but in recent decades, many workers in the eld have come to dislikethisterm, feelingthatitpromisestoomuch, and/orthatitsuggestsanemphasis onuniversal constructions. Universal constructions are a majorthemeofthiscourse,buttheyarenotallthattheeldisabout.The most popular replacement term is General Algebra, and I have used itin the title of these notes; but it has the disadvantage that in some contexts,itmaynotbeunderstoodasreferringtoaspecicarea.Below,ImostlysayGeneralAlgebra,butoccasionallyrefertotheolderterm.1.6. OtherreadingAsidefromthesenotes,thereisnorecommendedreadingforthecourse,butI will mention here some items in the list of references that you might like tolookat. Thebooks[1], [6], [7], [12], [20] and[22] areothergeneral textsinGeneral (a.k.a. Universal)Algebra. Of these, [12] isthemosttechnical andencyclopedic.[20]and[22]areboth,likethesenotes,aimedatstudentsnotnecessarilyhavingadvancedprior mathematical background; however [22]diersfromthiscourseinemphasizingpartial algebras. [7] hasincommonwiththis course the viewpoint that this subject is animportant tool foralgebraistsofall sorts, anditgivessomeinterestingapplicationstogroups,divisionrings,etc..[31] and[33] are standardtexts for Berkeleys basic graduate algebracourse. (Some subset of Chapters 2-7 of the present notes can, incidentally, beusefulsupplementaryreadingforstudentstakingsuchacourse.)Thoughwe6 1 Aboutthecourse,andthesenoteswill not assume the full material of such a course (let alone the full contents ofthose books), you may nd them useful references. [33] is more complete andrigorous; [31] is sometimes better on motivation. [25]-[27] include similar ma-terial.Apresentationofthecorematerialofsuchacourseatapproximatelyanhonorsundergraduatelevel, withdetailedexplanationsandexamples, is[28].Each of [6], [7], [12], [20], [22] and [26] gives a little of the theory of lattices,introducedinChapter6ofthesenotes.Extensivetreatmentsofthissubjectcanbefoundin[4]and[13].Chapter7ofthesenotesintroducescategorytheory.[8]isthepaperthatcreatedthatdiscipline, andstill verystimulatingreading; [19] isageneraltext onthesubject. [10] deals withanimportant areaof categorytheorythatourcourseleavesout.Forthethought-provokingpaperfromwhichtheideaswedevelopinChapter10come,see[11].Anamusingparodyof someof thematerial weshall beseeinginChap-ters5-10is[18].1.7. Numeration; history; advice; webaccess; requestforcorrectionsThese notes are divided into chapters, and each chapter into sections. In eachsection,Iusetwonumberingsystems:onethatembraceslemmas,theorems,denitions, numbereddisplays, etc.; theotherforexercises. Thenumberofeachitembeginswiththechapter-and-sectionnumbers. Thisisfollowedbya.andthe number of the result, display, etc., or a:andthe num-ber of the exercise. For instance, insectionm.n, i.e., sectionn of Chap-term,wemighthavedisplay(m.n.1),followedbyDenitionm.n.2,followedbyTheoremm.n.3, andinterspersedamongthese, Exercisesm.n :1, m.n :2,m.n :3, etc.. Thereasonforusingacommonnumberingsystemforresults,denitions, anddisplaysisthatitiseasiertondProposition3.2.9if itisbetween Lemma 3.2.8 and display (3.2.10) than it would be if it were Propo-sition 3.2.3, located between Lemma 3.2.5 and display (3.2.1). The exercisesform a separate system. There is a List of Exercises at the end of these notes,withtelegraphicdescriptionsoftheirsubjects.These notes began around 1971, as mimeographed outlines of my lectures,which I handed out to the class, and gradually improved in successive teach-ingsof thecourse. Withtheadventof computerword-processing, suchim-provements became mucheasier tomake, andthe outline evolvedintoareadable development of the material. In Spring and Summer 1995 they werepublishedbytheshort-livedBerkeleyLectureNotesseries, andforseveralyears after that, by my late colleague Henry Helson. I have continued to revisethemeachtimeItaughtthecourse;Ineverstopndingpointsthatcallfor1.8 Acknowledgements 7improvement. But hopefully, they are now in a state that justies publicationinbookform.In recentdecades, I have kept the notes available online, as an alternativeto buying a paper copy. Though this will not be feasible for the nal publishedversion, IintendtokeeptheversionthatIsubmittotheSpringereditorialstaavailableviamywebsitehttp://math.berkeley.edu/~gbergman .To other instructors who may teach from these notes (and myself, in case Iforget),I recommendmoving quite fastthroughthe easyearly material,andmuch more slowly toward the end, where there are more concepts new to thestudents, andmorenontrivial proofs. Roughlyspeaking, thehardmaterialbegins with Chapter 8. A ner description of the hard parts would be: 7.9-7.11, 8.3, 8.9-8.12, 9.9-9.10, and Chapter 10. However, this judgement is basedonteachingthecoursetostudentsmostof whomhaverelativelyadvancedbackgrounds. Forstudentswhohavenotseenordinalsorcategoriesbefore(the kind I had in mind in writing these notes), the latter halves of Chapters 5and7wouldalsobeplacestomoveslowly.The last two sections of each of Chapters 7, 8 and 9 are sketchy (to varyingdegrees),sostudentsshouldbeexpectedeithertoreadthemmainlyfortheideas,ortoputinextraeorttoworkoutdetails.After many years of editing, reworking, and extending these notes, I knowone reason why the copy-from-the-blackboard system has not been generallyreplacedbythedistributionofmaterialinwrittenform:Agoodsetofnotestakesanenormousamountoftimetodevelop. ButIthinkthatitisworththeeort.Commentsandsuggestionsonanyaspectofthesenotesorganizational,mathematical or other, includingindications of typographical errors arewelcomeatgbergman@math.berkeley.edu .1.8. AcknowledgementsThoughIhadpickedupsomecategorytheoryhereandthere, therstex-tensivedevelopmentof itthatIreadwasMacLane[19], andmuchof thematerial on categories in these notes ultimately derives from that book. I cannolongerreconstructwhichcategory-theoretictopicsIknewbeforereading[19], but mydebt tothat workis considerable. Cohns book[7] was simi-larlymyrstexposuretoasystematicdevelopmentofGeneralAlgebra;andFreydsfascinatingpaper[11] isthesourceof thebeautiful resultof 10.4,whichI considerthe climaxofthecourse.I amalso indebtedto morepeoplethanI cannamefor helpwithspecicquestions inareas wheremyback-groundhadgaps.Forthedevelopmentandmaintenanceofthelocallyenhancedversionofthetext-formattingprogramtroff, inwhichIpreparedearlierversionsof8 1 Aboutthecourse,andthesenotesthesenotes, I amindebtedtoEdMoy, toFranRizzardi, andtoD. MarkAbrahams, andforhelpwiththeconversiontoLATEX, toGeorgeGratzer,PaulVojta,and,especially,ArturoMagidin.Finally, I amgrateful tothemanystudents whohavepointedout cor-rections tothesenotes over theyears inparticular, toArturoMagidin,DavidWasserman, MarkDavis, JosephFlenner, BorisBukh, ChrisCulter,andLynnScow.PartI.Motivationandexamples.Inthe next three chapters, we shall lookat particular cases of algebraicstructures and universal constructions involving them, so as to get some senseofthegeneralresultswewillwanttoproveinthechaptersthatfollow.Theconstructionoffreegroupswillbeourrstexample.WeprepareforitinChapter2bymakingprecisesomeconceptssuchasthatof agroup-theoreticexpressioninaset of symbols; then, inChapter 3, weconstructfreegroups byseveral mutuallycomplementaryapproaches. InChapter 4,welookatalargenumberofotherconstructionsfromgrouptheory,semi-grouptheory,ringtheory,etc.whichhave,togreaterorlesserdegrees,thesamespiritasthefreegroupconstruction,andalso,forvariety,attwosuchconstructionsfromtopology.9Chapter2Makingsomethingsprecise2.1. GeneralitiesMostnotationwillbeexplainedasitisintroduced.Iwillassumefamiliaritywith basic set-theoretic and logical notation: for for all (universal quan-tication), for there exists (existential quantication), for and, andforor. Functionswill beindicatedbyarrows, whiletheirbehavioronelementswill beshownbyat-tailedarrows, . Thatis, if afunctionXY carries anelement x toanelement y, this maybesymbolizedx y ( xgoesto y ).If S isasetand anequivalencerelationonS,thesetofequivalenceclassesunderthisrelationwillbedenotedS/.We will (with rare exceptions, which will be noted) write functions on theleft of their arguments, i.e., f(x) rather thanxf, and understand compositefunctions fg tobedenedsothat (fg)(x) = f(g(x)).2.2. Whatisagroup?Looselyspeaking,agroupisaset Ggivenwithacomposition(ormultipli-cation,orgroupoperation) :GG G, aninverseoperation :G G,and a neutral elemente G, satisfying certain well-known laws. (We will sayneutral element rather than identity element to avoid confusion with theotherimportantmeaningof thewordidentity, namelyanequationthatholdsidentically.)Themostconvenientwaytomakeprecisethisideaofasetgivenwiththreeoperationsistodenethegrouptobe,nottheset G, butthe4-tuple(G,,,e). Infact, fromnowon, alettersuchas Grepresentingagroupwillstandforsucha4-tuple,andtherstcomponent,calledtheunderlyingsetofthegroup,willbewritten [G[. ThusG=([G[,,,e).102.2 Whatisagroup? 11Forsimplicity, manymathematiciansignorethisformal distinction, andusealetter suchas Gtorepresent bothagroupandits underlyingset,writing x G, for instance, wheretheymean x [G[. This is okay, aslongas one always understands what precise statement suchashorthandstatementstandsfor.Notethattobeentirelyprecise,if GandHaretwogroups, weshouldusedierentsymbols, say Gand H, Gand H, eGandeH, fortheoperationsof GandH. Howpreciseandformaloneneedstobedependsonthesituation. Sincetheaimof thiscourseistoabstracttheconceptof algebraicstructureandstudywhatmakesthesethingstick,weshallbesomewhatmorepreciseherethaninanordinaryalgebracourse.(Many workers in General Algebra use a special type-font, e.g., boldface, torepresentalgebraicobjects,andregulartypefortheirunderlyingsets.Thus,wherewewillwrite G = ([G[,,,e), theymightwrite G = (G,,,e).)Perhapstheeasiestexerciseinthecourseis:Exercise2.2:1. GiveaprecisedenitionofahomomorphismfromagroupGtoagroup H, distinguishingbetweentheoperations of Gandtheoperationsof H.We will often refer to a homomorphismf : G Has a map fromGtoH. That is, unless the contrary is mentioned, maps between mathematicalobjectsmeanmapsbetweentheirunderlyingsetswhichrespecttheirstruc-ture. Notethatifwewishtorefertoasetmapnotassumedtorespectthegroupoperations,wecancallthisamapfrom [G[ to [H[ .Theuseofletters (and) fortheoperationsofagroup,andthefunc-tional notation (x,y), (z) whichthis entails, are desirable for statingresultsinaformwhichwillgeneralizetoawideclassofothersortsofstruc-tures. But when actually working with elements of a group, we will generallyuseconventional notation, writing xy (or xy, or sometimes, inabeliangroups, x+y) for (x,y), andz1(or z) for (z). When we do this, wemayeithercontinuetowrite G = ([G[,,,e), orwrite G = ([G[, , 1,e).Let us nowrecall the conditions whichmust be satisedbya4-tupleG = ([G[, , 1,e), where [G[ isaset,isamap [G[ [G[ [G[, 1is a map [G[ [G[, andeis an element of [G[, for Gto be called a group:(2.2.1)( x, y, z [G[) (xy)z =x(yz),( x [G[) ex=x=xe,( x [G[) x1 x=e=xx1.Thereisanotherdenitionofgroupthatyouhaveprobablyalsoseen:Ineect,agroupisdenedtobeapair ([G[, ), suchthat [G[ isaset,and isamap [G[ [G[ [G[ satisfying12 2 Makingsomethingsprecise(2.2.2)( x, y, z [G[) (xy)z= x(yz),( e [G[) (( x [G[) ex = x = xe) (( x [G[) ( y [G[) yx = e = xy).It is easy to show that given([G[, ) satisfying (2.2.2), there exist a uniqueoperation1and a unique elementesuch that([G[, , 1,e)satises (2.2.1) remember the standard results saying that neutral elements and 2-sided in-versesareuniquewhentheyexist. Thus, theversions(2.2.1)and(2.2.2)oftheconcept of agroupprovideequivalent information. Our descriptionofgroupsas4-tuplesmaythereforeseemuneconomicalcomparedwithoneusingpairs, but we will stickwithit. We shall eventuallysee that, moreimportant thanthe number of terms inthe tuple, is the fact that condi-tion(2.2.1)consistsofidentities,i.e.,universallyquantiedequations,while(2.2.2) doesnot. But wewill at timesacknowledgetheideaof theseconddenition; forinstance, whenweask(imprecisely)whethersomesemigroupisagroup.Exercise2.2:2. (i) If Gisagroup,letusdeneanoperationGon [G[byG(x,y) = xy1. Doesthepair G

= ([G[,G) determinethegroup([G[, , 1,e) ?(I.e., if G1and G2yieldthesamepair, G

1=G

2, mustG1= G2 ?Somestudentshaveaskedwhetherby =Iheremean=.No,Imean =.)(ii) Suppose [X[ isanysetand : [X[[X[ [X[ anymap. Canyouwritedownasetofaxiomsforthepair X= ([X[, ), whichwillbeneces-saryandsucientforittoarisefromagroupGinthemannerdescribedabove?(Thatis,assuming [X[ and given,trytondconvenientneces-saryandsucientconditionsfortheretoexistagroup Gsuchthat G

,denedasin(i),isprecisely([X[,).)If yougetsuchasetof axioms, thentrytoseehowbrief andsimpleyoucanmakeit.Idontknowthefullanswerstothefollowingvariantquestion:Exercise2.2:3. Againlet Gbeagroup, andnowdene G(x,y) =x y1 x. Consider thesamequestionsfor ([G[,G) that wereraisedfor([G[,G) intheprecedingexercise.My point in discussing the distinction between a group and its underlyingset, and between groups described using (2.2.1) and using (2.2.2), was not tobepetty,buttomakeusconsciousofthevariouswaysweusemathematicallanguage so that we can use it without its leading us astray. At times we willbowtoconvenienceratherthantryingtobeconsistent. Forinstance, sincewedistinguishbetweenagroupanditsunderlyingset, weshouldlogicallydistinguishbetweentheset of integers, theadditivegroupof integers, themultiplicativesemigroupof integers, theringof integers, etc.; butweshallinfactwriteall of these Zunlessthereisareal dangerof ambiguity, oraneedtoemphasizeadistinction. Whenthereissuchaneed, wecanwrite(Z,+, ,0)= Zadd, (Z, ,1)= Zmult, (Z,+, , ,0,1)= Zring, etc.. We2.4 Arity 13maylikewiseusereadymadesymbolsforsomeotherobjects,suchas eforthetrivialsubgroupofagroupG, ratherthaninterruptingadiscussionto set up a notation that distinguishes this subgroup from its underlying set.The approach of regarding sets with operations as tuples, whose rst mem-beristhesetandwhoseothermembersaretheoperations, applies, aswehavejustnoted, tootheralgebraicstructuresthangroupstosemigroups,rings, lattices, and the more exotic beasties we will meet on our travels. To beabletodiscussthegeneralcase,wemustmakesureweareclearaboutwhatwemeanbysuchconceptsasn-tupleofelementsandn-aryoperation.Weshallreviewtheseinthenexttwosections.2.3. IndexedsetsIf I and Xaresets, anI-tupleof elementsof X, orafamilyof elementsof Xindexedby I willbedenedformallyasafunctionfromI toX, butweshall writeit (xi)iIratherthan f : I X. Thedierenceisoneofviewpoint. Wethinkof suchfamiliesasarraysof elementsof X, whichwekeep track of with the help of an index setI, while when we writef : A B,wearemost ofteninterestedinsomeproperties relatinganelement of Aandits imagein B. But thedistinctionis not sharp. Sometimes thereisaninterestingfunctional relationbetweentheindices i andthevalues xi;sometimes typographical or other reasons dictate the use of x(i) rather thanxi.Therewill beaminorformal exceptiontotheabovedenitionwhenwespeak of an n-tuple of elements of X (n 0). In these beginning chapters, Iwill take this to mean a function from 1,. . . ,ntoX, written(x1,. . . ,xn)or (xi)i=1,..., n, despitethefactthatsettheoristsdenethenaturalnumbernrecursively to be the set 0,. . . ,n1. Most set theorists, for consistencywith that denition, write their n-tuples (x0,. . . ,xn1); and we shall switchtothatnotationafterreviewingthesettheoristsapproachtothenaturalnumbersinChapter5.If I and Xaresets, thenthesetof all functionsfromI to X, equiv-alently, ofall I-tuplesofmembersof X, iswritten XI. Likewise, Xnwilldenotethesetof n-tuplesof elementsof X, denedasaboveforthetimebeing.2.4. ArityAn n-ary operation on a set Smeans a mapf : Sn S. For n = 1, 2, 3thewords are unary, binary, and ternary respectively. If f is an n-ary operation,14 2 Makingsomethingsprecisewe call nthe arity of f. More generally, given any set I, an I-ary operationonSisdenedasamapSI S.Thus, thedenitionofagroupinvolvesonebinaryoperation, oneunaryoperation,andonedistinguishedelement,orconstant, e. Likewise,aringcan be described as a 6-tupleR = ([R[,+, , ,0,1), where+andare bi-nary operations on [R[, is a unary operation, and0,1are distinguishedelements,satisfyingcertainidentities.Onemaymakethesedescriptionsmorehomogeneousinformbytreatingdistinguishedelementsas0-aryoperationsofouralgebraicstructures.In-deed,sinceann-aryoperationonS issomethingthatturnsoutavalueinSwhen we feed innarguments inS, it makes sense that a 0-ary operationshould be something that gives a value inSwithout our feeding it anything.Or, looking at it formally, S0is the set of all maps from the empty set toS,ofwhichthereisexactlyone;so S0isaone-elementset,soamapS0 Sdetermines,andisdeterminedby,asingleelementof S.We note also that distinguished elements show the right numerical behav-ior to be called zeroary operations. Indeed, if f andgare an m-ary and ann-aryoperationonS, andi a positive integer m, thenoninsertingg inthe i-th place of f, we get an operationf( , . . . , , g( ,. . . , ), , . . . , )of aritym+n 1. Now if, instead, g is an element of S, then when we putitintothei-thplaceof f weget f(, . . . , ,g, , . . . , ), an(m 1)-aryoperation,asweshouldif g isthoughtofasanoperationofarityn = 0.Strictlyspeaking, elementsandzeroaryoperationsareinone-to-onecor-respondence rather than being the same thing: one must distinguish betweenamapS0 S, andits(unique)valueinS. Butsincetheygiveequivalentinformation,wecanchoosebetweentheminsettingupourdenitions.Soweshallhenceforthtreatdistinguishedelementsinthedenitionofgroups, rings, etc., aszeroaryoperations, andwewill ndthattheycanbehandled essentially like the other operations. I say essentially because therearesomeminorwaysinwhichzeroaryoperationsdierfromoperationsofpositivearity. Most notably, onthe emptyset X= , thereis auniquen-aryoperationfor eachpositive n, but nozeroaryoperation. Sometimesthistrivialfactwillmakeadierenceinanargument.2.5. Group-theoretictermsOne is often interested in talking about what relations hold among the mem-bersofoneoranothertupleofelementsofagrouporotheralgebraicstruc-ture.Forexample,everypairofelements (,) ofagroupsatisestherela-tion ()1=1 1. Someparticularpair (,) ofelementsofsomegroupmaysatisfytherelation= 2.Ingeneral,agroup-theoreticrelationinafamilyofelements (i)IofagroupGmeans an equationp(i) = q(i) holding inG, wherepandq are2.5 Group-theoreticterms 15expressions formed from an I-tuple of symbols using formal group operations,1ande. So to studyrelations ingroups,we needto dene the setofallformal expressionsintheelementsofaset Xundersymbolicoperationsofmultiplication,inverseandneutralelement.Thetechnical wordforsuchaformal expressionisaterm. Intuitively,agroup-theoretic termis aset of instructions onhowtoapplythegroupoperations to a family of elements. E.g., starting with a set of three symbols,X= x,y,z, anexampleof agroup-theoretictermin Xisthesymbol(y x) (y1); or we might write it ((y,x),(y)). Whichever waywewriteit, theideais: applytheoperation tothepair (y,x), applytheoperation to the element y, and then apply the operationto the pair ofelementssoobtained,takeninthatorder.Theideacanberealizedwhenwearegivenamapf oftheset Xintotheunderlyingset [G[ ofagroupG=([G[,G, G,eG), say x , y , z (,, [G[). Wecanthen dene the result of evaluating the term((y,x),(y)) using the mapf astheelement G(G(,),G()) [G[, thatis, ()(1).Letustrytomaketheconceptofgroup-theoretictermprecise.Thesetof all termsintheelementsof X, underformal operations,1and e shouldbeaset T= TX, ,1, ewiththefollowingproperties:(aX) Foreveryx X, T containsasymbolrepresenting x.(a.) Forevery s,t T, T containsasymboliccombinationof s andtunder.(a1) Forevery s T, T containsanelementgottenbysymbolicappli-cationof1to s .(ae) T containsanelementsymbolizing e.(b) Eachelementof T canbewritteninoneandonlyonewayasoneandonlyoneofthefollowing:(bX) Thesymbolrepresentinganelementof X.(b.) Thesymboliccombinationoftwomembersof T under .(b1) The symbol representing the result of applying1to an elementof T.(be) Thesymbolrepresenting e.(c) Every element of T can be obtained fromthe elements of X viathegivensymbolicoperations. Thatis, T hasnopropersubsetsatisfying(aX)(ae).In functional language, (aX) says that we are to be given a functionX T(the symbol for x function); (a.)says we have another function,whichwecall formal product, fromTT to T; (a1)givesafunction TT,the formal inverse, and (ae) a distinguished element of T. Translating ourdenitionintothislanguage,wegetDenition2.5.1. Bythesetofall termsintheelementsof Xundertheformalgroupoperations , , eweshallmeanaset T whichis:16 2 Makingsomethingsprecise(a) givenwithfunctionssymbT:X T, T:T2 T, T:T T, and eT:T0 T,suchthat(b) eachofthesemapsisone-to-one,theirimagesaredisjoint,andT istheunionofthoseimages,and(c) T isgeneratedby symbT(X) undertheoperations T, T, and eT;i.e., ithasnopropersubsetwhichcontains symbT(X) andisclosedunderthoseoperations.The next exercise justies the use of the word the in the above denition.Exercise2.5:1. AssumingT andT

are two sets given with functions thatsatisfyDenition2.5.1,establishanaturalone-to-onecorrespondencebe-tweentheelements of T and T

. (Youmust, of course, showthat thecorrespondenceyousetupiswell-dened, andisabijection. Suggestion:Let Y = (symbT(x),symbT (x)) [ x X TT

, andlet F betheclosureof Y undercomponentwiseapplicationof , ande. Showthat F isthegraphofabijection.Whatpropertieswillcharacterizethisbijection?)Exercise2.5:2. Is condition(c) of Denition2.5.1aconsequence of (a)and(b)?Howcanweobtainaset T withthepropertiesof theabovedenition?Oneapproachistoconstructelementsof T asnitestringsofsymbolsfromsomealphabetwhichcontainssymbolsrepresentingtheelementsof X, ad-ditionalsymbols (or), (or1),ande, andperhapssomesymbolsofpunctuation. Butweneedtobecareful. Forinstance, if wedened Ttotake a string of symbols sand a string of symbols t to the string of symbolsst, and Ttotakeastringof symbols s tothestringof symbols s1,then condition (b) would not be satised! For a string of symbols of the formx yz (wherex, y, z X) could be obtained by formal multiplication eitherof xandyz, or of x yandz. In other words, Ttakes the pairs (x,yz)and(x y,z) to the same string of symbols, so it is not one-to-one. Likewise,the expressionx y1could be obtained either as T(x,y1) or as T(x y),sotheimagesof Tand Tarenotdisjoint. (Ithappensthatintherstcase, the two interpretations of x yz come to the same thing in any group,because of the associative law, while in the second, the two interpretations donot: (1) and()1are generally distinct for elements , of a groupG. Butthepointisthatinbothcasescondition(b)fails,makingtheseex-pressions ambiguous as instructions for applying group operations. Note thatanotational systeminwhich xyz wasambiguousintheabovewaycouldneverbeusedinwritingdowntheassociativelaw; andwritingdownidentitiesisoneoftheuseswewillwanttomakeoftheseexpressions.)On the other hand, it is not hard to show that by introducing parenthesesamong our symbols, and lettingT(s,t) be the string of symbols (s t), and2.5 Group-theoreticterms 17T(s) thestringofsymbols (s1), wecangetasetofexpressionssatisfyingtheconditionsofourdenition.Exercise2.5:3. Verifytheaboveassertion.(How,precisely,willyoudeneT ?Whatassumptionsmustyoumakeonthesetofsymbolsrepresentingelementsof X ?Doyouallowsomeelements symbT(x) tobestringsofothersymbols?)Anothersymbolismthatwillworkistodenethevalueof Tat s andt tobethestringof symbols (s,t), andthevalueof Tat s tobethestringofsymbols (s).Exercise2.5:4. Assumingtheelements symbT(x) aredistinctsinglechar-acters, andthat , and e are distinct characters distinct fromthecharacters symbT(x), letusdenethevalueof Tonelements s andtto be the symbol st, and the value of Tonsto be the symbol s. WilltheresultingsetofstringsofsymbolssatisfyDenition2.5.1?Though the strings-of-symbols approach can be extended to other kinds ofalgebraswithnitaryoperations,suchasrings,lattices,etc.,adisadvantageof that method is that one cannot, in any obvious way, use it for algebras withoperationsof innitearities. Evenif oneallowsinnitestringsof symbols,indexed by the natural numbers or the integers, one cannot string two or moresuch innite strings together to get another string of the same sort. One can,however, for aninniteset I, createI-tuples whichhaveI-tuples amongtheirmembers, andthisleadstothemoreversatileset-theoreticapproach.Letusshowitforthecaseofgroup-theoreticterms.Chooseanysetoffourelements, whichwill bedenoted , ,1and e.For eachx X, denesymbT(x) to be the ordered pair(,x); fors,t T,dene T(s,t) to be the ordered 3-tuple (,s,t); for s T dene T(s) tobe the ordered pair (1,s), and nally, dene eTto be the 1-tuple (e). LetT bethesmallestsetclosedundertheaboveoperations.Nowitisabasiclemmaofsettheorythatnoelementcanbewrittenasann-tupleinmorethanoneway; i.e., if (x1, . . . , xn)=(x

1, . . . , x

n ), thenn

=nand xi=x

i(i=1, . . . ,n). ItiseasytodeducefromthisthattheaboveconstructionwillsatisfytheconditionsofDenition2.5.1.Exercise2.5:5. Would there have been anything wrong with deningsymbT(x) =x insteadof (,x) ?If so, canyoundawaytomodifythe denitions of Tetc., so that the denitionsymbT(x) = xcan alwaysbeused?I leave it to you to decide (or not to decide) which construction for group-theoretic terms you prefer to assume during these introductory chapters. Weshall onlyneedtheproperties giveninDenition2.5.1. Fromnowon, weshall oftenuseconventional notationforsuchterms, e.g., (xy)(x1). Inparticular,weshalloftenidentifyXwithitsimage symbT(X) TX,,1, e.Wewill usethemoreformal notationof Denition2.5.1mainlywhenwe18 2 Makingsomethingsprecisewanttoemphasizeparticulardistinctions, suchasthatbetweentheformaloperations Tetc.,andtheoperations Getc.ofaparticulargroup.2.6. EvaluationNowsuppose Gisagroup,andf :X [G[ asetmap,inotherwords,anX-tupleofelementsof G. GivenaterminanX-tupleofsymbols,s T =TX,,1, ewe wish to say how to evaluatesat this familyf of elements, so as to get avaluesf [G[. We shall do this inductively (or more precisely, recursively;wewilllearnthedistinctionin 5.3).If s=symbT(x) forsome x Xwedene sf=f(x). If s=T(t,u),thenassuminginductivelythat wehavealreadydened tf,uf [G[, wedene sf=G(tf,uf). Likewise, if s=T(t), weassumeinductivelythattfis dened, and denesf= G(tf). Finally, for s = eTwe denesf= eG.Sinceeveryelement s T isobtainedfromsymbT(X) bytheoperationsT, T, eT, andinauniquemanner, thisconstructiongivesoneandonlyonevalue sfforeachs.Wehavenotdiscussedthegeneral principlesthatallowonetomakere-cursivedenitions liketheabove. Weshall developtheseinChapter 5, inpreparationforChapter9wherewewill dorigorouslyandinfull general-itywhatwearesketchinghere.Somestudentsmightwanttolookintothisquestionforthemselvesatthispoint,soIwillmakethis:Exercise2.6:1. Show rigorously that the procedure loosely described aboveyields a unique well-dened mapT [G[. (Suggestion: Adapt the methodsuggestedforExercise2.5:1.)Intheabovediscussionofevaluation,wexedf [G[X, andgotafunc-tion T [G[, takingeach s T to sf [G[. Ifwevary f aswell as T,wegetatwo-variableevaluationmap,(TX,,1, e) [G[X [G[,takingeachpair (s,f) to sf. Finally, wemightxan s T, anddeneamap sG : [G[X [G[ by sG(f)=sf(f [G[X); thisrepresentssub-stitutioninto s.Forexample,suppose X= x,y,z, letusidentify [G[Xwith[G[3, andlet s betheterm(yx)(y1) T. ThenforeachgroupG, sGis the operation taking each 3-tuple(,,) of elements of Gto theelement ( ) 1G. Suchoperationswill beof importancetous, sowegivethemaname.2.7 Termsinotherfamiliesofoperations 19Denition2.6.1. Let Gbe a group andna nonnegative integer. Let T=Tn,1, , edenotetheset of group-theoreticterms in n symbols. Thenforeachs T, we will let sG : [G[n [G[ denote the map taking each n-tuplef [G[ntotheelement sf [G[. Then-aryoperations sGobtainedinthiswayfromterms s T willbecalledthederivedn-aryoperationsof G.(Someauthorscallthesetermoperations.)Notethatdistincttermscaninducethesamederivedoperation.E.g.,theassociativelawforgroupssaysthatforanygroup G, thederivedternaryoperationsinducedbytheterms (xy)z and x(yz) arethesame. Asanotherexample,intheparticulargroupS3(thesymmetricgrouponthreeelements),thederivedbinaryoperationsinducedbytheterms (xx)(yy)and(yy)(xx) arethesame,thoughthisisnottrueinallgroups.(Itistrueinalldihedralgroups.)Someotherexamplesofderivedoperationsongroupsarethebinaryop-eration of conjugation, commonly written= 1 (induced by the termy1 (xy)), the binary commutator operation, [,] =11 , theunaryoperationof squaring, 2=, andthetwobinaryoperations andof Exercises 2.2:2 and 2.2:3. Some trivial examples are also important:theprimitivegroupoperationsgroupmultiplication, inverse, andneutralelementarebydenitionalsoderivedoperations;andnally,onehasverytrivialderivedoperationssuchastheternarysecondcomponentfunction,p3, 2(,,) =, inducedby yT{x, y, z},1, , e. (Here p3, 2stands forprojectionof3-tuplestotheirsecondcomponent.)2.7. TermsinotherfamiliesofoperationsTheaboveapproachcanbeappliedtomoregeneralsortsofalgebraicstruc-tures. Let beanorderedpair ([[,ari), where[[ isasetof symbols(thoughtofasrepresentingoperations),andari isafunctionassociatingtoeach [[ anonnegativeinteger ari(), theintendedarityof (2.4).(For instance, in the group case which we have been considering, we have eec-tively taken [[ = ,,e, ari() = 2, ari() = 1, ari(e) = 0. Incidentally,thecommonestsymbol,amongspecialists,forthearityofanoperationisn(), butIwill use ari() toavoidconfusionwithotherusesofthelettern.) Thenan-algebrawillmeanasystemA = ([A[,(A)||), where [A[isaset,andforeach [[, Aissomeari()-aryoperationon [A[ :A: [A[ari() [A[.Foranyset X, wecannowmimictheprecedingdevelopmenttogetasetT= TX, , thesetoftermsinelementsof Xundertheoperationsof ;andgivenany-algebraA, wecangetsubstitutionandevaluationmapsasbefore,andsodenederivedoperationsof A.20 2 MakingsomethingspreciseThelong-rangegoalofthiscourseistostudyalgebras Ainthisgeneralsense. In order to discover what kinds of results we want to prove about them,weshall devoteChapters3and4tolookingatspecicsituationsinvolvingfamiliar sorts of algebras. But let megivehereafewexercises concerningthesegeneralconcepts.Exercise2.7:1. Onthe set 0,1, let M3denote the ternary major-ity vote operation; i.e., for a, b, c 0, 1, let M3(a,b,c) be 0 iftwo or more of a, b and c are 0, or 1 if two or more of themare 1. One canformvarious terms inasymbolic operation M3(e.g.,p(w,x,y,z) = M3(x,M3(z,w,y),z)) andthenevaluatetheseintheal-gebra (0,1,M3) togetoperationson 0,1 derivedfromM3.General problem: Determinewhichoperations(of arbitraryarity)on0,1 canbeexpressedasderivedoperationsofthisalgebra.Asstepstowardansweringthisquestion, youmighttrytodeterminewhethereachofthefollowingcanorcannotbesoexpressed:(a) The5-arymajorityvotefunction M5 : 0,15 0,1, denedintheobviousmanner.(b) The binary operationsup . (I.e., sup(a,b) = 0if a = b = 0; otherwisesup(a,b) = 1.)(c) Theunaryreversaloperationr, denedbyr(0) = 1, r(1) = 0.(d) The 4-aryoperation N4, describedas the majorityvote function,wheretherstvoterhasextratie-breakingpower;i.e., N4(a,b,c,d) =the majority value amonga, b, c, dif there is one, while if a+b +c +d = 2weset N4(a,b,c,d) = a.Advice: (i) If yousucceedinprovingthat some operation s is notderivable fromM3, try to abstract your argument by establishing a generalpropertythat all operations derivedfromM3must have, but which sclearlydoesnothave.(ii) Amistakesomestudentsmakeistothinkthataformulasuchas s(,)=M3(0,,) denesaderivedoperation. Butsinceoursystem(0,1,M3) doesnot includethezeroaryoperation 0(nor 1), M3(0,x,y) isnotaterm.Exercise2.7:2. (Question raised by Jan Mycielski, letter of Jan. 17, 1983.)LetCdenotethesetofcomplexnumbers,andexptheexponentialfunc-tionexp(x) = ex, aunaryoperationonC.(i) Doesthealgebra (C,+, ,exp) haveanyautomorphismsotherthantheidentityandcomplexconjugation?(Anautomorphismmeansabijec-tion of the underlying set with itself, which respects the operations.) I dontknowtheanswertothisquestion.It is not hard to prove using the theory of transcendence bases of elds([31, VI.1], [33, VIII.1]) that theautomorphismgroupof (C,+, ) isinnite(cf.[31,ExerciseVI.6(b)],[33,ExerciseVIII.1]).Acoupleofeasyresultsintheoppositedirection,whichyoumayproveandhandin,are(ii) Thealgebra (C,+, ) hasnocontinuousautomorphismsotherthanthetwomentionedin(i).(iii) If we write cj for the unary operation of complex conjugation, thenthealgebra (C,+, ,cj) hasnoautomorphismsotherthanidandcj.(iv) A mapC Cis an automorphism of (C,+, ,exp) if andonlyifitisanautomorphismof (C,+,exp).2.7 Termsinotherfamiliesofoperations 21Exercise2.7:3. Givenoperations 1,. . . ,r(of variousnitearities)onaniteset S, andanotheroperation on S, describeatestthatwilldetermineinanitenumberofstepswhether isaderivedoperationof1,. . . ,r.Thearitiesconsideredsofarhavebeennite; thenextexercisewill dealwith terms in operations of possibly innite arities. To make this reasonable,let us note some naturally arising examples of operations of countably innitearityonfamiliarsets:Onthereal unitinterval [0,1] :(a) theoperationlimsup(limitsuperior),denedbylimsupixi=limisupjixj,(b) theoperationdenedbys(a1,a2,. . . ) =

2iai.Onthesetofreal numbers 1 :(c) the continued fraction operation, c(a1,a2,. . . ) = a1 +1/(a2 +1/(. . . )).Ontheclassofsubsetsofthesetofintegers:(d) theoperation

ai,(e) theoperation

ai.Exercise2.7:4. Suppose isapair ([[,ari), where[[ isagainasetof operationsymbols, butwherethearities ari() maynowbeniteorinnite cardinals; andlet X be aset of variable-symbols. Suppose wecanformaset T of terms satisfyingthe analogs of conditions (a)-(c)of Denition2.5.1. For s, t T, letuswrite s >

t if t isimmediatelyinvolvedin s, thatis, if s hastheform(u1,u2,. . . ) where [[,andui= t forsome i.(i) Showthatifallthearities ari() arenite,thenforeachterms wecan nd a nite boundB(s) on the lengthsnof sequencess1,. . . ,sn Tsuchthat s = s1>

. . . >

sn.(ii) If not all ari() are nite, andXis nonempty, show that there existterms s forwhichnosuchniteboundexists.(iii) Inthe situationof (ii), is it possible tohave aright-innite chains = s1>

. . . >

sn>

. . . inT ?(iv) Showthatonecannothaveacycle s1>

. . . >

sn>

s1inT.Until we come to Chapter 9, we shall rarely use the word algebra in thegeneral sense of this section. But the reader consulting the index should keepthissenseinmind, sinceitisusedtherewithreferencetogeneral conceptsofwhichwewillbeconsideringspeciccasesintheinterveningchapters.Chapter3FreegroupsInthischapter,weintroducetheideaofuniversalconstructionsthroughtheparticular case of free groups. We shall rst motivate the free group concept,thendevelopthreewaysofconstructingsuchgroups.3.1. MotivationSuppose Gisagroupandwetake(say)threeelements a,b,c [G[, andconsiderwhatgroup-theoreticrelationsthesesatisfy. Thatis, letting T bethe set of all group-theoretic terms in three symbols x, y andz, we look atpairsofelements p(x,y,z), q(x,y,z) T, andif pG(a,b,c) = qG(a,b,c)in [G[, wesaythat (a,b,c) satisestherelationp = q. Wenote:Lemma3.1.1. Suppose F and Garegroups,suchthat F isgeneratedbythree elements a,b,c [F[, while ,, are anythree elements of G.Thenthefollowingconditionsareequivalent:(a) Everygroup-theoreticrelation p=q satisedby (a,b,c) in F isalsosatisedby (,,) inG.(b) Thereexistsagrouphomomorphismh: FGunderwhich a ,b , c .Further, when these conditions hold, the homomorphism h of (b) isunique.If theassumptionthat a, b and c generate F isdropped, onestill has(b) = (a).Proof. Notyetassumingthat a, b and c generate F, suppose hisaho-momorphismasin(b).ThenIclaimthatforall p T,h(pF(a,b,c)) =pG(,,).223.1 Motivation 23Indeed,thesetof p T forwhichtheaboveequationholdsiseasilyseentocontain x, y and z, andtobeclosedundertheoperationsof T, henceitisallof T. Statement(a)follows,givingthenalassertionofthelemma.If,further, a, b andc generate F, theneveryelementof [F[ canbewrittenpF(a,b,c) for some p, sotheaboveformulashows that givensuch a, band c, thehomomorphismh isdeterminedby , and , yieldingthenext-to-lastassertion.Finally, suppose a, b and c generate F and(a) holds. For each g=pF(a,b,c) [F[, dene h(g) = pG(,,). Toshowthatthisgivesawell-denedmapfrom[F[ to[G[, notethatif wehavetwowaysof writinganelement g [F[, say pF(a,b,c)=g=qF(a,b,c), thentherelationp=qissatisedby(a,b,c) inF, henceby(a),itissatisedby(,,) inG,hencethetwovaluesourdenitionprescribesfor h(g), namelypG(,,)andqG(,,), arethesame.That this set map is a homomorphismfollows fromthe way evalua-tionof group-theoretictermsisdened. Forinstance, given g [F[, sup-posewewant toshowthat h(g1) =h(g)1. Wewrite g =pF(a,b,c).Then (T(p))F(a,b,c) = g1, so our denition of h gives h(g1) =(T(p))G(,,) =pG(,,)1=h(g)1. Thesamereasoningappliestoproductsandtotheneutralelement. .Exercise3.1:1. Showbyexamplethat if a,b,c does not generate F,thencondition(a) of the above lemmacanholdand(b) fail, andalsothat(b)canholdbut hnotbeunique.(Youmayreplace (a,b,c) withasmallerfamily, (a,b) or (a), ifyoulike.)Lemma 3.1.1 leads one to wonder: Among all groups F given with gener-ating 3-tuples of elements(a,b,c), is there one in which these three elementssatisfy the smallest possible set of relations? We note what the above lemmawouldimplyforsuchagroup:Corollary3.1.2. Let F beagroup, and a,b,c [F[. Thenthefollowingconditionsareequivalent:(a) a,b,c generate F, andtheonlyrelationssatisedby a,b,c in F arethose relations satised by every 3-tuple (,,) of elements in everygroupG.(b) ForeverygroupG, andevery3-tupleofelements (,,) inG, thereexistsauniquehomomorphismh: F Gsuchthat h(a)=, h(b)=,h(c) = . .Only one point in the deduction of this corollary from Lemma 3.1.1 is notcompletelyobvious;Iwillmakeitanexercise:Exercise3.1:2. In the situation of the above corollary, show that (b) impliesthata, bandcgenerateF. (Hint: LetGbe the subgroup of Fgeneratedbythosethreeelements.)24 3 FreegroupsIve been speaking of 3-tuples of elements for concreteness; the same obser-vationsarevalidforn-tuplesforany n, andgenerally,forX-tuplesforanyset X. AnX-tupleofelementsof F meansasetmapX [F[, sointhisgeneral context, condition(b)abovetakestheformgivenbythenextde-nition.(Butmakingthisdenitiondoesnotanswerthequestionofwhethersuchobjectsexist!)Denition3.1.3. Let Xbe a set. By a free groupFon the set X, we shallmean a pair (F,u), where F is a group, andua set mapX [F[, havingthefollowinguniversal property:For every groupG, and every set mapv : X [G[, there exists a uniquehomomorphismh: F Gsuch thatv= hu; i.e., making the diagram belowcommute.X [F[u`````` v[G[F1 hG(Intheabovediagram, therstvertical arrowalsorepresentsthehomo-morphismh, regardedasamapontheunderlyingsetsofthegroups.)Corollary 3.1.2 (as generalized to X-tuples) says that(F,u)is a free grouponXifandonlyiftheelements u(x) (x X) generate F, andsatisfynorelations except those that hold for every X-tuple of elements in every group.Inthis situation, onesaysthat theseelementsfreelygenerate F, hencethetermfreegroup. Notethatif suchan F exists, thenbydenition, anyX-tupleof membersof anygroup Gcanbeobtained, inauniqueway, astheimage,underagrouphomomorphismF G, oftheparticularX-tupleu. HencethatX-tuplecanbethoughtof asauniversal X-tupleof groupelements,sothepropertycharacterizingitiscalledauniversalproperty.Wenoteafewelementaryfacts andconventions about suchobjects. If(F,u) is afree groupon X, thenthe map u: X [F[ is one-to-one.(Thisiseasytoprovefromtheuniversalproperty,plusthewell-knownfactthatthereexistgroupswithmorethanoneelement. Thestudentwhohasnotseenfreegroupsdevelopedbeforeshouldthinkthisargumentthrough.)Hencegivenafreegroup, it iseasytoget fromit onesuchthat themapu isactuallyaninclusion X [F[. Hencefornotational convenience, onefrequently assumes that this is so; or, what is approximately the same thing,oneoftenusesthesamesymbolforanelementof Xanditsimagein [F[.If (F,u) and(F

,u

) are both free groups on the same set X, there is auniqueisomorphismbetweenthemasfreegroups, i.e., respectingthemaps3.2 Thelogiciansapproach:constructionfromterms 25uandu

. (Cf.diagrambelow.)X

u``````u

[F[[F

[`FF

`(If youhavent seenthisresult before, againseewhetheryoucanworkout the details. For the technique youmight lookaheadto the proof ofProposition4.3.3.)Asanytwofreegroupson Xarethusessentiallythesame,onesometimesspeaksofthefreegrouponX.OnealsooftensaysthatagroupF isfreetomeanthereexistssomeset Xandsomemapu:X [F[ suchthat (F,u) isafreegrouponX. Whenthisholds, Xcanalwaysbetakentobeasubsetof[F[, andutheinclusionmap.But it is time we proved that free groups exist. We will show three dierentwaysofconstructingtheminthenextthreesections.Exercise3.1:3. SupposeonereplacesthewordgroupbynitegroupthroughoutDenition3.1.3.Showthatforanynonemptyset X, nonitegroupexistshavingthestateduniversalproperty.3.2. Thelogicians approach: constructionfromgroup-theoretictermsWe knowfromCorollary 3.1.2 that if a free group F on three genera-tors a,b,c exists, theneachof itselementscanbewritten pF(a,b,c) forsomegroup-theoretictermp, andthattwosuchelements, pF(a,b,c) andqF(a,b,c), areequalifandonlyiftheequation p=q issatisedbyev-erythreeelementsofeverygroup, i.e., followsfromthegroupaxioms. Thissuggeststhatwemaybeabletoconstructsuchagroupbytakingthesetof all group-theoretictermsinthreevariables, constructinganequivalencerelation p q onthis set whichmeans theequalityof p and q is aconsequenceofthegroupaxioms,takingfor [F[ thequotientofoursetofterms by this relation, and dening operations ,1ande on [F[ in somenaturalmanner.Thisweshallnowdo!Let Xbe any set, andT= TX, ,1, ethe set of all group-theoretic termsin the elements of X. What conditions must a relation satisfy for p q26 3 Freegroupstobethecondition pv=qv forsomemap v of Xintosomegroup G?Well,thegroupaxiomstellusthatitmustsatisfy( p,q,r T) (pq)r p(qr), (3.2.1)( p T) (pe p) (ep p), (3.2.2)( p T) (pp1 e) (p1 p e). (3.2.3)Also,justthewell-denednessoftheoperationsof Gtellsusthat( p,p

,q T) (p p

) =((pq p

q) (qp qp

)), (3.2.4)( p,p

T) (p p

) =(p1 p1). (3.2.5)Finally,ofcourse, mustbeanequivalencerelation:( p T) p p, (3.2.6)( p,q T) (p q) =(q p), (3.2.7)( p,q,r T) ((p q) (q r)) =(p r). (3.2.8)So let us take for the least binary relation onT satisfying conditions(3.2.1)-(3.2.8).Let us note what this means, andwhyit exists: Recall that abinaryrelationonaset T isformallyasubset R T T; whenwewrite pq,this is understoodtobe anabbreviationfor (p,q) R. Leastmeanssmallest with respect to set-theoretic inclusion. Our conditions (3.2.1)-(3.2.8)are in the nature of closure conditions, and, as with all sets dened by closureconditions,theexistenceofaleastsetsatisfyingthemcanbeestablishedintwoways:Wemaycapturethissetfromabovebyformingtheintersectionofallbinary relations onT satisfying (3.2.1)-(3.2.8) the set-theoretic intersectionof these relations as subsets of T T. (Note, incidentally, that if we think ofsuch relations as predicates rather than as sets, this intersection

becomesa(generallyinnite)conjunction_.) Thekeypointtoobserveisthateachof these conditions is such that an intersection of relations satisfying it againsatises it. Hence the intersection of all relations satisfying (3.2.1)-(3.2.8) willbetheleastsuchrelation.Orwecanbuilditupfrombelow. Let R0denotetheemptyrelation T T, and recursively construct the i +1-st relationRi+1from the i-th,byadding toRithose elements thatconditions (3.2.1)-(3.2.8)saymust alsobeinR, giventhattheelementsof Riarethere.Precisely,weletRi+1=Ri(eltsalreadyconstructed) ((pq)r, p(qr)) [ p,q,r T (eltsarisingby(3.2.1)) ... ... (p,r) [ ( q) (p,q) Ri (q,r) Ri. (eltsarisingby(3.2.8))3.2 Thelogiciansapproach:constructionfromterms 27Wenowdene R=

iRi. Itisstraightforwardtoshowthat Rsatises(3.2.1)-(3.2.8),andthatanysubsetof TT satisfying(3.2.1)-(3.2.8)mustcontainR; soR, looked at as a binary relation onT, is the desired leastrelation.By(3.2.6)-(3.2.8), isanequivalencerelation; solet [F[ =T/, thesetofequivalenceclassesofthisrelation;i.e.,writing[p] fortheequivalenceclassof p T, [F[ = [p] [ p T. WemapXinto [F[ bythefunctionu(x) =[x](or, if wedonotidentify symbT(x) with x inourconstructionof T, byu(x) = [symbT(x)]). Wenowdeneoperations , 1ande on [F[ by[p][q] =[pq], (3.2.9)[p]1=[p1], (3.2.10)e =[e]. (3.2.11)That the rst two of these are well-dened follows respectively from prop-erties(3.2.4)and(3.2.5)of! (Withthethirdthereisnoproblem.)Fromproperties (3.2.1)-(3.2.3) of , it follows that ([F[, , 1,e) satises thegroupaxioms.E.g.,given[p], [q], [r] [F[, ifweevaluate ([p][q])[r] and[p]([q][r]) in[F[, weget [(pq)r] and [p(qr)] respectively, whichareequalby(3.2.1).WritingF forthegroup([F[, , 1,e), itisclearfromour construction of that every relation satised by the images inF of theelements of Xis a consequence of the group axioms; so by Corollary 3.1.2 (orrather, the generalization of that corollary with X-tuples in place of 3-tuples),F hasthedesireduniversalproperty.Toseethisuniversalpropertymoredirectly,suppose v isanymapX [G[, where Gisagroup. Write p vq tomean pv=qvin G. Clearlythe relation vsatises conditions (3.2.1)-(3.2.8), hence it contains the leastsuchrelation, our . Soawell-denedmap h: [F[ [G[ is givenbyh([p]) =pv [G[, andit follows fromthewaytheoperations of F, andtheevaluationof terms in Gat theX-tuple v, aredened, that h is ahomomorphism, andistheuniquehomomorphismsuchthat hu=v. ThuswehaveProposition3.2.12. (F,u), constructedas above, is afree grouponthegivenset X. .Soafreegrouponeveryset Xdoesindeedexist!Remark3.2.13. Thereisaviewpointthatgoesalongwiththisconstruction,whichwill behelpful inthinkingabout universal constructionsingeneral.Supposethatwearegivenaset X, andthatweknowthat Gisagroup,withamap v : X [G[. Howmuchcanwesayabout Gfromthisfact28 3 Freegroupsalone?Wecannamecertainelementsof G, namelythe v(x) (x X), andalltheelementsthatcanbeobtainedfromthesebythegroupoperationsofG(e.g., (v(x) v(y))1 ((v(y)1 e)1 v(z)) if x,y,z X). A particular Gmay contain more elements than those obtained in such ways, but we have nowayofgettingourhandsonthemfromthegiveninformation. Wecanalsoderive fromthe identities for groups certainrelations that these elementssatisfy,(e.g., (v(x)v(y))1= v(y)1 v(x)1). Theelements v(x) may,inparticular cases, satisfy more relations than these, but again we have no wayof deducing these additional relations. If we now gather together this limiteddatathatwehaveaboutsuchagroupGthequotientofasetoflabelsforcertainelementsbyasetof identicationsamongthesewendthatthiscollectionof dataitselfformsagroupwithamapof Xintoit; andis,infact,auniversalsuchgroup!Remark3.2.14. At the beginning of this section, I motivated our constructionbysayingthat shouldmeanequalitythatfollowsfromthegroupax-ioms. I then wrote down a series of eight rules, (3.2.1)-(3.2.8), all of which areclearly valid procedures for deducing equations which hold in all groups. Whatwasnotobviouswaswhethertheywouldbesucienttoyieldallsuchequa-tions. Buttheyweretheproof of thepuddingbeingthat (T/, , 1,e)wasshowntobeagroup.This is anexampleof averygeneral typeof situationinmathematics:Some class, in this case, a class of pairs of group-theoretic terms, is describedfromabove, i.e., isdenedastheclassof all elementssatisfyingcertainrestrictions(inthiscase, thosepairs (p,q) TT suchthattherelationp=q holdsonall X-tuplesof elements of all groups). Weseekawayofdescribingitfrombelow,i.e.,ofconstructingorgeneratingallmembersoftheclass.Someprocedurewhichproducesmembersofthesetisfound,andone seeks to show that this procedure yields the whole set or, if it does not,oneseekstoextendittoaprocedurethatdoes.The inverse situation is equally important, where we are given a construc-tion which builds up a set, and we seek a convenient way ofcharacterizingtheelementsthatresult.Exercise2.7:1wasofthatform.Youwillseemoreexamples of bothsituations throughout this course, and, infact, inmosteverymathematicscourseyoutake.Exercise3.2:1. Provedirectlyfrom(3.2.1)-(3.2.8)thatfor x,y X, (x y)1 y1 x1. (Yoursolutionshouldshowexplicitlyeachapplicationyoumakeofeachofthoseconditions.)Exercise3.2:2. Does the relation of the preceding exercise followfrom(3.2.1)-(3.2.3)and(3.2.6)-(3.2.8)alone?Notethatinourrecursiveconstructionoftheset R(thatis,therelation), repeated application of (3.2.1)-(3.2.3) was really unnecessary; these con-ditions give the same elements of Reach time they are applied, so we might3.3 Freegroupsassubgroupsofbigenoughdirectproducts 29as well just have applied them the rst time, and only applied (3.2.4)-(3.2.8)afterthat.Lessobviousistheanswerto:Exercise3.2:3. (A. Tourubaro) Canthe constructionof R be done inthree stages: First take the set P of elements givenby(3.2.1)-(3.2.3),thenformtheclosure Qof thissetunderapplicationsof (3.2.4)-(3.2.5)(as before, byrecursionor as anintersection), andnally, obtain Rasthe closure of Qunder applications of (3.2.6)-(3.2.8) (another recursion orintersection)? This procedure will yield some subset of TT; the questioniswhetheritisthe Rwewant.Whatif wedothingsinadierentorderrst(3.2.1)-(3.2.3), then(3.2.6)-(3.2.8),then(3.2.4)-(3.2.5)?3.3. Freegroupsassubgroupsof bigenoughdirectproductsAnotherwayofgettingagroupinwhichsomeX-tupleofelementssatisesthe smallest possible set of relations is suggested by the following observation.Let G1andG2betwogroups,andsupposewearegivenelements1,1,1 [G1[, 2,2,2 [G2[.TheninthedirectproductgroupG = G1 G2wehavetheelementsa=(1,2), b=(1,2), c=(1,2),and we nd that the set of relations satised bya,b,c inGis precisely theintersectionofthesetofrelationssatisedby1,1,1inG1andthesetof relationssatisedby 2,2,2in G2. Thismaybeseenfromthefactthatforanys T,sG(a,b,c) =(sG1(1, 1, 1), sG2(2, 2, 2)),asiseasilyveriedbyinduction.Moregenerally, if wetakeanarbitraryfamilyofgroups (Gi)iI, andineachGithree elements i,i,i, then in the product groupG = Gi, wecandenetheelementsa=(i)iI, b=(i)iI, c=(i)iI,andtherelationsthatthesesatisfywill bejustthoserelationssatisedsi-multaneouslybyour3-tuplesinallofthesegroups.Thissuggeststhatbyusinga largeenough suchfamily,we couldarriveatagroupwiththreeelements a,b,c whichsatisfyasmallest possiblesetofrelations.30 3 FreegroupsHowlargeafamily(Gi,i,i,i) shouldweuse?Well,wecouldbesureofgettingtheleastsetofrelationsifwecouldusetheclassof all groupsandall 3-tuplesof elementsof these. Buttakingthedirectproductofsuchafamilywouldgiveusset-theoreticindigestion.WecancutdownthissurfeitofgroupsabitbynotingthatforanygroupGiandthreeelements i,i,i, if welet Hidenotethesubgroupof Gigenerated by these three elements, it will suce for our product to involve thegroup Hi, ratherthanthewholegroup Gi, sincetherelationssatisedbyi, iandiin the whole groupGiand in the subgroupHiare the same.Nowanitelygeneratedgroupis countable (meaningnite or countablyinnite), so we see that it would be enough to let (Gi,i,i,i) range overallcountablegroups,andall3-tuplesofelementsthereof.However, theclassofall countablegroupsisstill notaset. Indeed, eventheclassof one-elementgroupsisnotaset, becausewegetadierent(inthe strict set-theoretic sense) group for each choice of that one element. (Forthosenotfamiliarwithsuchconsiderations: Insettheory, everyelementofasetisaset.Ifwehadasetofall one-elementgroups,thenwecouldformfromthisthesetofallmembersoftheirunderlyingsets,whichwouldbethesetof all sets; andoneknowsthatthisdoesnotexist.)Butthisisclearlyjustaquibbleobviously, if wechooseanyone-elementsetx, andtaketheuniquegroupwiththisunderlyingset, itwill serveaswell asanyotherone-elementgroupsofarashonestgroup-theoreticpurposesareconcerned.Inthesameway,Iclaimwecanndagenuinesetofcountablegroupsthatuptoisomorphismcontains all thecountablegroups. Namely, let S beaxed countably innite set. Then we can form the set of all groups Gwhoseunderlying sets [G[ are subsets of S. Or, to hit more precisely what we want,let(3.3.1) (Gi,i,i,i) [ i Ibethesetof all 4-tuplessuchthat Giisagroupwith[Gi[ S, and i,iandiaremembersof [Gi[. NowforanycountablegroupHandthreeelements ,, [H[, wecanclearlyndanisomorphism fromoneofthesegroups, say Gj(j I), to H, suchthat (j) =, (j) =,(j) = ; so(3.3.1)isbigenoughforourpurpose.Sotaking(3.3.1)asabove,let P bethedirectproductgroupIGi, leta,b,c betheI-tuples (i), (i), (i) [P[, andlet F bethesubgroupofP generatedbya, b andc. Iclaimthat F isafreegroupona, b andc.Wecouldprovethisbyconsideringthesetofrelationssatisedby a, b,c inF as suggested above, but let us instead verify directly that F satisestheuniversal propertycharacterizingfreegroups(Denition3.1.3). Let Gbe anygroup, and ,, three elements of G. We want toprove thatthereexistsauniquehomomorphismh: F Gcarrying a,b,c [F[ to,, [G[ respectively.UniquenesswillbenoproblembyconstructionFis generated bya, bandc, so if such a homomorphism exists it is unique.3.3 Freegroupsassubgroupsofbigenoughdirectproducts 31Toshowtheexistenceof h, notethatthesubgroup Hof Ggeneratedby,, iscountable, henceaswehavenoted, thereexistsforsome j Ianisomorphism: Gj =Hcarrying j,j,j [Gj[ to ,, [H[.Nowtheprojectionmap pjof theproductgroup P= Giontoitsj-thcoordinate takes a, b andc toj,j,j, hence composing this projectionwith , wegetahomomorphismh: FGhavingthedesiredeectona,b,c, asshowninthediagrambelow.F a,b,c [F[ P = Gia,b,c [Gi[ pjGjj,j,j [Gj[H ,, [H[ G ,, [G[Forausefulwaytopicturethisconstruction,thinkof P asthegroupofall functions on the base-space I, taking at each point i I a value inGi:, , , ,, , ,, ,,, ,Gjj,, , , , , , ,, ,groups Gi`` Ielementsof P=IGiThenF isthesubgroupoffunctionsgeneratedby a, b andc. Nowgiven,, in any groupG, identify the subgroup of Gthat they generate with32 3 Freegroupsan appropriateGj(j I). Then the homomorphismhthat we constructedabovemaybethoughtof astakingeachelementof F toitsvalueatthepoint j. Wehavechosenourspace I andvaluesfor a, b andc sucientlyeclectically so that it is possible to choose points at whicha, bandctake on(uptoisomorphism)any3-tupleofvaluesinanygroup.Thus,thefunctionsa, b andc areauniversal3-tupleofgroup-elements.The same argument works if we replace 3-tuple by X-tuple, where Xisanycountableset.Hereweusetheobservationthatagroupgeneratedbyacountablefamilyofelementsiscountable.For Xofarbitrarycardinality,onecaneasilyshowthatagroup HgeneratedbyanX-tupleof elementshascardinality max(card(X), 0). Henceweget:Proposition3.3.2. Let X be any set. Take a set S of cardinalitymax(card(X), 0), andlet (Gi,ui) [ i I betheset of all pairs suchthat Giis agroupwith[Gi[ S, and uiis amap X [Gi[ (i.e., anX-tupleofelementsof Gi). Let P=IGi, andmap Xinto P byden-ing u(x) (x X) tobetheelement withcomponent ui(x) at each i. LetF bethesubgroupof P generatedby u(x) [ x X.Thenthepair (F,u) isafreegroupontheset X. .Digression: Let S3bethesymmetricgrouponthreeletters. Supposewehadbeguntheaboveinvestigationwithalessambitiousgoal:merelytondagroupJ withthreeelements a,b,c suchthat(3.3.3)For every choice of three elements, , [S3[, thereexistsauniquehomo-morphismh: J S3takinga,b,cto,,, respectively.J a,b,c [J[ hS3, , [S3[Thenwecouldhaveperformedtheaboveconstructionjust using4-tuples(S3, , , ) (, , [S3[) as our (Gi,i,i,i). There are 63=216such4-tuples, so P wouldbethedirectproductof 216copiesof S3, anda,b,c wouldbeelementsof thisproductwhich, asonerunsoverthe216coordinates, take on all possible combinations of values inS3. The subgroupJ theygeneratewouldindeedsatisfy(3.3.3).Thisleadsto:Exercise3.3:1. Doescondition(3.3.3)characterize (J,a,b,c) uptoiso-morphism?Ifnot,istheresomeadditionalconditionthat (J,a,b,c) sat-iseswhichtogetherwith(3.3.3)determinesituptoisomorphism?Exercise3.3:2. InvestigatethestructureofthegroupJ, andmoregener-ally, of the analogous groups constructed fromS3using dierent numbersof generators. To make the problem concrete, try to determine, or estimateaswell aspossible, theordersof thesegroups, for 1,2,3 andgenerally,for ngenerators.Themethodsbywhichwehaveconstructedfreegroupsinthisandtheprecedingsectiongoover essentiallyword-for-wordwithgroupreplaced3.3 Freegroupsassubgroupsofbigenoughdirectproducts 33byring, lattice, oragreatmanyothertypesof mathematical objects.The determination of just what classes of algebraic structures admit this andrelatedsortsof universal constructionsisoneof thethemesof thiscourse.Thenextexerciseconcernsanegativeexample.Exercise3.3:3. Statewhatwouldbemeantbyafreeeldonaset X ,and show that no such object exists for any setX. If one attempts to applythetwomethodsofthisandtheprecedingsectiontoprovetheexistenceoffreeelds,wheredoeseachofthemfail?Exercise3.3:4. LetZ[x1,. . . ,xn] bethepolynomialringinnindetermi-natesovertheintegers ( =thefreecommutativeringon ngeneratorscf. 4.12below).Itseldoffractions(x1,. . . ,xn), theeldofrationalfunctionsinnindeterminatesovertherationals,looksinsomewayslikeafreeeldon ngenerators. E.g., oneoftenspeaksof evaluatingara-tional functionatsomesetof valuesof thevariables. Cansomeconceptof freeeldbesetup, perhapsbasedonamodieduniversal property,oronsomeconceptofcomparingrelationsintheeldoperationssatisedbyn-tuplesofelementsintwoelds, intermsofwhich(x1,. . . ,xn) isindeedthefreeeldonngenerators?Exercise3.3:5. A division ring (or skew eld or seld) is a ring (associativebut not necessarily commutative) in which every nonzero element is invert-ible.Ifyoundasatisfactoryanswertotheprecedingexercise,youmightconsiderthequestionofwhetherthereexistsinthesamesenseafreedivi-sionring onngenerators. (This was a longstanding open question, whichwas nally answered in 1966, and then again, by a very dierent approach,about ve years later. I can refer interested students to papers in this area.)Therearemanyhybrids andvariants of thetwoconstructions wehavegivenforfreegroups. Forinstance, wemightstartwiththeset T oftermsinX, anddene p q (for p,q T) tomeanthatforeverymapv of Xintoagroup G, onehas pv=qvin G. Nowforeachpair (p,q) TTsuchthat p q fails tohold, we canchoose amap up, qof X intoagroup Gp, qsuchthat pup, q ,=qup, q. WecanthenformthedirectproductgroupP= Gp, q, taketheinducedmapu:X [P[, andcheckthatthesubgroup F generatedbytheimageof thismapwill satisfycondition(a)of Corollary3.1.2. Interestingly, for Xcountable, thisconstructionusesaproductoffewergroups Gp, qthanweusedintheversiongivenabove.Finally, considerthe following construction,whichsuers from severe set-theoreticdiculties,butisstillinteresting.(Iwonttrytoresolvethesedif-cultieshere,butwilltalksloppily,asthoughtheydidnotoccur.)Deneageneralizedgroup-theoreticoperationinthreevariablesasanyfunctionp whichassociatestoeverygroupGandthreeelements , , [G[ anelement p(G,,,) [G[. Wecanmultiplytwosuchoperationsp andq bydening(pq)(G, ,,) =p(G,,,)q(G,,,) [G[.34 3 Freegroupsfor all groups Gandelements , , [G[. Wecansimilarlydenethemultiplicativeinverseofsuchanoperationp, andtheconstantoperatione.Weseethattheclassof generalizedgroup-theoreticoperationswill satisfythegroupaxiomsundertheabovethreeoperations.Nowconsiderthethreegeneralizedgroup-theoreticoperations a, b andc denedbya(G,,,) = , b(G,,,) = , c(G,,,) = .Letusdeneaderivedgeneralizedgroup-theoreticoperationasoneob-tainable froma, b andc by the operations of product, inverse, and neutralelementdenedabove. Thenthesetof derivedgeneralizedgroup-theoreticoperations will formafreegrouponthegenerators a, b and c. (This isreallyjustadisguisedformofournaivedirectproductofallgroupsidea.)Exercise3.3:6. Call a generalized group-theoretic operationpfunctorial iffor every homomorphism of groupsf : G H, one hasf(p(G,,,)) =p(H, f(), f(), f()) (,, [G[). (Wewill seethereasonfor thisnameinChapter7.)Showthatallderivedgroup-theoreticoperationsarefunctorial.Istheconversetrue?Exercise3.3:7. Samequestionforfunctorialgeneralizedoperationsontheclassofallnitegroups.3.4. Theclassicalconstruction:freegroupsasgroupsofwordsTheconstructionsdiscussedabovehavethedisadvantageofnotgivingveryexplicit descriptions of freegroups. Weknowthat everyelement of afreegroup F ontheset Xarisesfromatermintheelementsof Xandthegroupoperations, but wedont knowhowtotell whether twosuchtermssay (b(a1b)1)(a1b) and e yieldthesameelement; inotherwords,whether ((1)1)(1) = e istrueforallelements , ofallgroups.Ifitis, thenbytheresultsof 3.2onecanobtainthisfactsomehowbytheprocedurescorrespondingtoconditions(3.2.1)-(3.2.8); if itisnot, thentheideasof 3.3suggestthatweshouldtrytoprovethisbylookingforsomeparticularelementsforwhichitfails, insomeparticulargroupinwhichweknowhowtocalculate.Buttheseapproachesarehit-and-miss.Inthissection, weshall constructthefreegroupon Xinamuchmoreexplicitway.Wewillthenbeabletoanswersuchquestionsbycalculatinginthefreegroup.Werstrecallanimportantconsequenceoftheassociativeidentity:thatproducts can be written without parentheses. For example, given elementsa,b,c ofagroup,theelements a(c(ab)), a((ca)b), (ac)(ab), (a(ca))b and3.4 Theclassicalconstruction:groupsofwords 35((ac)a)b are all equal. It is conventional, and usually convenient, to say, Letus therefore write their common value asa c a b. However, we will soon wanttorelatetheseexpressionstogroup-theoreticterms; soinsteadof droppingparentheses, let us agree totake a(c(ab)) as the commonformtowhichweshall reducetheaboveveexpressions, andgenerally, let us notethatanyproductof elementscanbereducedbytheassociativelawtoonewithparenthesesclusteredtotheright: xn (xn1(. . . (x2x1) . . . )).Inparticular,giventwoelementswritteninthisform,wecanwritedowntheir product and reduce it to this form by repeatedly applying the associativelaw:(3.4.1)(xn (... (x2x1) . . . ))(ym (... (y2y1) . . . ))=xn (... (x2(x1 (ym (... (y2y1) . . . )))) . . . ).Ifwewanttondtheinverseofanelementwritteninthisform,wemayusetheformula (xy)1= y1x1, anotherconsequenceofthegrouplaws.Byinductionthis gives (xn( . . . (x2x1) . . . ))1=( . . . (x11x12) . . . ) x1n,whichwemayreduce,againbyassociativity,to x11( ... (x1n1x1n) . . . ).Moregenerally,ifwestartedwithanexpressionoftheformx1n( ... (x12x11) . . . ),where eachfactor is either xior x1i, andthe exponents are indepen-dent, thenthe above methodtogether withthe fact (x1)1=x (an-other consequence of the groupaxioms) allows us towrite its inverse asx11( . . . (x1n1x1n) . . . ), whichis of thesameformas theexpressionwestartedwith;justas(3.4.1)showsthattheproductoftwoexpressionsoftheaboveformreducestoanexpressionofthesameform.Notefurtherthatiftwosuccessivefactors x1iandx1i+1arerespectivelyxandx1forsomeelement x, orarerespectivelyx1andxforsome x,thenbythegroupaxiomsoninversesandtheneutral element(andagain,associativity), wecandropthis pair of factors unless theyaretheonlyfactorsintheproduct,inwhichcasewecanrewritetheproductas e.Finally, easy consequences of the group axioms tell us what the inverse of eis (namelye), and how to multiply anything bye. Putting these observationstogether, weconcludethatgivenanyset Xof elementsof agroup G, thesetofelementsof Gthatcanbewritteninoneoftheforms(3.4.2)e or x1n(... (x12x11) . . . ),where n 1, each xi X, andnotwosuccessivefactorsareanelementof Xandtheinverseofthesameelement,ineitherorder,36 3 Freegroupsis closed under products and inverses. So this set must be the whole subgroupof Ggenerated byX. In other words, any member of the subgroup generatedbyXcanbereducedbythegroupoperationstoanexpression(3.4.2).In the preceding paragraph, Xwas a subset of a group. Now let Xbe anarbitraryset,andasin 3.2,let T bethesetofallgroup-theoretictermsinelementsof X(Denition2.5.1). Forconvenience, letusassume T chosensoas tocontain X, with symbTbeingtheinclusionmap. (If youprefernot tomakethis assumption, thenintheargument tofollow, youshouldinsert symbTatappropriatepoints.)Let Tred T (redstandingforreduced) denotetheset of terms of theform(3.4.2). If s,t Tred, wecanformtheirproduct st inT, andthen,aswehavejustseen,rearrangeparenthesestogetanelementof Tredwhichisequivalentto st sofarasevaluationatX-tuplesof elementsof groupsisconcerned. Letuscall thiselementst. Thus, st has the properties that it belongs toTred, and thatfor any mapv : X [G[ (Ga group) one has (s t)v= (st)v. In the sameway, given s Tred, wecanobtainfroms1T anelementweshall calls() Tred, suchthatforanymapv :X [G[, onehas (s1)v= (s())v.Areanyfurtherreductionspossible?ForaparticularX-tupleofelementsof aparticulargrouptheremaybeequalitiesamongthevaluesof dierentexpressionsoftheform(3.4.2);butweareonlyinterestedinreductionsthatcan be done in all groups. No more are obvious; but can we be sure that somesneakyapplicationofthegroupaxiomswouldntallowustoprovesometwodistinct terms (3.4.2) to have the same evaluations at all X-tuples of elementsofallgroups?(Insuchacase,weshouldnotlosehope,butshouldintroducefurtherreductionsthatwouldalwaysreplaceoneoftheseexpressionsbytheother.)Letusformalizetheconsequencesof theprecedingobservations, andin-dicatethesignicanceofthequestionwehaveasked.Lemma3.4.3. Foreach s T, thereexistsan s

Tred(i.e., anelementof T ofoneoftheformsshownin(3.4.2))suchthat(3.4.4) foreverymap v of Xintoanygroup G, sv= s

vin [G[.Moreover,ifoneofthefollowingstatementsistrue,all are:(a) Foreach s T, thereexistsaunique s

Tredsatisfying(3.4.4).(b) If s, t aredistinct elementsof Tred, then s=t isnotanidentityforgroups;thatis,forsome Gandsome v :X [G[, sv ,= tv.(c) The4-tuple F= (Tred, ,(),eT) isagroup.(d) The4-tuple F= (Tred, ,(),eT) isafreegrouponX.Proof. Wegettherstsentenceofthelemmabyaninduction,whichIwillsketch briey. The assertion holds for elementsx X: we simply takex

= x.Nowsupposeittruefortwoterms s,t T. Toestablishitfor st T,dene (st)

=s

t

. Onelikewisegetsitfor s1using s

(), anditis3.4 Theclassicalconstruction:groupsofwords 37clear for e. It follows from condition (c) of the denition of group-theoreticterm(Denition2.5.1)thatitistrueforallelementsof T.Theequivalenceof (a) and(b) is straightforward. Assumingthesecon-ditions, let us verifythat the 4-tuple F denedin(c) is a group. Takep,q,r Tred. Then p (q r) and (p q) r aretwoelementsof Tred,callthems andt. Foranyv :X [G[, sv= tvbytheassociativelawforG. Hence by (b), s = t, proving that is associative. The other group lawsfor F arededucedinthesameway.Conversely,assuming(c),weclaimthatfordistinctelements s,t Tred,we can prove, as required for (b), that the equation s = t is not an identitybygettingacounterexampletothatequationinthisverygroupF. Indeed,ifwelet v betheinclusionX Tred= [F[, wecancheckbyinductiononnin(3.4.2)thatforall s Tred, sv=s. Hence s ,=t implies sv ,=tv, asdesired.Since(d)certainlyentails(c),ourproofwillbecompleteifwecanshow,assuming(c),that F hastheuniversalpropertyofafreegroup.Givenanygroup Gandmap v : X [G[, wemap[F[ =Tredto[G[ by s sv.From the properties of and(), we know that this is a homomorphismhsuchthat h[X(therestrictionof hto X) is v; andsince Xgenerates F,histheuniquehomomorphismwiththisproperty,asdesired. .Wellarestatements(a)-(d)true,ornot??Theusualwaytoanswerthisquestionistotestcondition(c)bywritingdownpreciselyhowtheoperations and()areperformed,andcheckingthegroupaxiomsonthem. Sinceatermoftheform(3.4.2)isuniquelyde-terminedbytheinteger n(whichwetaketobe 0 fortheterme) andthen-tupleofelementsof Xandtheirinverses, (x1n,. . . ,x11), onedescribesand()asoperationsonsuchn-tuples. E.g., onemultipliestwotuples(w,. . . ,x) and(y,. . . ,z) (where each of w,. . . ,z is an element of Xor asymbolic inverse of such an element) by uniting them as(w,. . . ,x,y,. . . ,z),thendroppingpairsoffactorsthatmaynowcancel(e.g., xandy aboveify is x1); and repeating this last step until no such cancelling pairs remain.But checking the associative law for this recursively dened operation turnsout to be very tedious, involving a number of dierent cases. (E.g., you mighttrycheckingassociativityfor (v,w,x)(x1,w1,y1)(y,w,z), andfor(v,w,x)(x1,z1,y1)(y,w,z), wherew, x, yandz are four distinctelementsof X. Bothcaseswork,buttheyaredierentcomputations.)Butthereisaneleganttrick, notaswell knownasitoughttobe, whichrescues us from the toils of this calculation. We construct a certainGwhichwe know to be a group, using which we can verify condition (b) rather thancondition(c)oftheabovelemma.Toseehowtoconstructthis G, letusgobacktobasicsandrecallwherethegroupidentities,whichweneedtoverify,comefrom.Theyareidenti-tieswhicharesatisedbypermutationsofanyset A, undertheoperationsofcomposingpermutations,invertingpermutations,andtakingtheidentity38 3 Freegroupspermutation. So let us try to describe a set Aon which the group we want toconstruct should act by permutations in as free a way as possible, specifyingthepermutationof Athatshouldrepresenttheimageofeachx X.Tostartourconstruction, let a beanysymbol notinx1[ x X.Nowdene Atobethesetofallstringsofsymbolsoftheform:(3.4.5)x1nx1n1. . . x11awhere n 0, each xi X, andnotwosuccessivefactorsx1iandx1i+1areanelementof Xandtheinverseofthatsameelement,ineitherorder.Inparticular,taking n = 0, weseethat a A.Let Gbethegroupofallpermutationsof A. Deneforeachx Xanelement v(x) [G[ asfollows.Givenb A,if b does not begin with the symbol x1, let v(x) take b tothesymbol xb, formedbyplacinganxatthebeginningofthesymbol b;if b doesbeginwithx1, sayb = x1c, let v(x) take b tothesymbol c, formedbyremoving x1fromthebeginningof b.Itisimmediatefromthedenitionof Athat v(x)(b) belongsto Aineachcase. To check thatv(x)is invertible, consider the map which sends a symbolxb to b, andasymbol c notbeginningwith x tothesymbol x1c; wendthatthisisa2-sidedinverseto v(x).Sowenowhaveamapv :X [G[. Asusual,thisinducesa