ABOOKOFABSTRACTALGEBRA

SecondEdition

CharlesC.PinterProfessorofMathematics

BucknellUniversity

DoverPublications,Inc.,Mineola,NewYork

Copyright

Copyright1982,1990byCharlesC.PinterAllrightsreserved.

BibliographicalNote

ThisDoveredition,firstpublishedin2010,isanunabridgedrepublicationofthe1990secondeditionoftheworkoriginallypublishedin1982bytheMcGraw-HillPublishingCompany,Inc.,NewYork.

LibraryofCongressCataloging-in-PublicationData

Pinter,CharlesC,1932Abookofabstractalgebra/CharlesC.Pinter.Dovered.

p.cm.Originallypublished:2nded.NewYork:McGraw-Hill,1990.Includesbibliographicalreferencesandindex.ISBN-13:978-0-486-47417-5ISBN-10:0-486-47417-81.Algebra,Abstract.I.Title.

QA162.P562010512.02dc22

2009026228

ManufacturedintheUnitedStatesbyCourierCorporation47417803

www.doverpublications.com

http://www.doverpublications.com

Tomywife,Donna,andmysons,

Nicholas,Marco,Andrs,andAdrian

CONTENTS*

Preface

Chapter 1 WhyAbstractAlgebra?HistoryofAlgebra.NewAlgebras.AlgebraicStructures.AxiomsandAxiomaticAlgebra.AbstractioninAlgebra.

Chapter 2 OperationsOperationsonaSet.PropertiesofOperations.

Chapter 3 TheDefinitionofGroupsGroups.ExamplesofInfiniteandFiniteGroups.ExamplesofAbelianandNonabelianGroups.GroupTables.TheoryofCoding:Maximum-LikelihoodDecoding.

Chapter 4 ElementaryPropertiesofGroupsUniquenessofIdentityandInverses.PropertiesofInverses.DirectProductofGroups.

Chapter 5 SubgroupsDefinitionofSubgroup.GeneratorsandDefiningRelations.CayleyDiagrams.CenterofaGroup.GroupCodes;HammingCode.

Chapter 6 FunctionsInjective,Surjective,BijectiveFunction.CompositeandInverseofFunctions.Finite-StateMachines.AutomataandTheirSemigroups.

Chapter 7 GroupsofPermutationsSymmetricGroups.DihedralGroups.AnApplicationofGroupstoAnthropology.

Chapter 8 PermutationsofaFiniteSetDecompositionofPermutationsintoCycles.Transpositions.EvenandOddPermutations.AlternatingGroups.

Chapter 9 IsomorphismTheConceptofIsomorphisminMathematics.IsomorphicandNonisomorphicGroups.CayleysTheorem.

GroupAutomorphisms.

Chapter 10 OrderofGroupElementsPowers/MultiplesofGroupElements.LawsofExponents.PropertiesoftheOrderofGroupElements.

Chapter 11 CyclicGroupsFiniteandInfiniteCyclicGroups.IsomorphismofCyclicGroups.SubgroupsofCyclicGroups.

Chapter 12 PartitionsandEquivalenceRelations

Chapter 13 CountingCosetsLagrangesTheoremandElementaryConsequences.SurveyofGroupsofOrder10.NumberofConjugateElements.GroupActingonaSet.

Chapter 14 HomomorphismsElementaryPropertiesofHomomorphisms.NormalSubgroups.KernelandRange.InnerDirectProducts.ConjugateSubgroups.

Chapter 15 QuotientGroupsQuotientGroupConstruction.ExamplesandApplications.TheClassEquation.InductionontheOrderofaGroup.

Chapter 16 TheFundamentalHomomorphismTheoremFundamentalHomomorphismTheoremandSomeConsequences.TheIsomorphismTheorems.TheCorrespondenceTheorem.CauchysTheorem.SylowSubgroups.SylowsTheorem.DecompositionTheoremforFiniteAbelianGroups.

Chapter 17 Rings:DefinitionsandElementaryPropertiesCommutativeRings.Unity.InvertiblesandZero-Divisors.IntegralDomain.Field.

Chapter 18 IdealsandHomomorphisms

Chapter 19 QuotientRingsConstructionofQuotientRings.Examples.FundamentalHomomorphismTheoremandSomeConsequences.PropertiesofPrimeandMaximalIdeals.

Chapter 20 IntegralDomainsCharacteristicofanIntegralDomain.PropertiesoftheCharacteristic.FiniteFields.ConstructionoftheFieldofQuotients.

Chapter 21 TheIntegersOrderedIntegralDomains.Well-ordering.Characterizationof UptoIsomorphism.MathematicalInduction.DivisionAlgorithm.

Chapter 22 FactoringintoPrimes

Idealsof .PropertiesoftheGCD.RelativelyPrimeIntegers.Primes.EuclidsLemma.UniqueFactorization.

Chapter 23 ElementsofNumberTheory(Optional)PropertiesofCongruence.TheoremsofFermtandEuler.SolutionsofLinearCongruences.ChineseRemainderTheorem.WilsonsTheoremandConsequences.QuadraticResidues.TheLegendreSymbol.PrimitiveRoots.

Chapter 24 RingsofPolynomialsMotivationandDefinitions.DomainofPolynomialsoveraField.DivisionAlgorithm.PolynomialsinSeveralVariables.FieldsofPolynomialQuotients.

Chapter 25 FactoringPolynomialsIdealsofF[x].PropertiesoftheGCD.IrreduciblePolynomials.Uniquefactorization.EuclideanAlgorithm.

Chapter 26 SubstitutioninPolynomialsRootsandFactors.PolynomialFunctions.Polynomialsover .EisensteinsIrreducibilityCriterion.PolynomialsovertheReals.PolynomialInterpolation.

Chapter 27 ExtensionsofFieldsAlgebraicandTranscendentalElements.TheMinimumPolynomial.BasicTheoremonFieldExtensions.

Chapter 28 VectorSpacesElementaryPropertiesofVectorSpaces.LinearIndependence.Basis.Dimension.LinearTransformations.

Chapter 29 DegreesofFieldExtensionsSimpleandIteratedExtensions.DegreeofanIteratedExtension.FieldsofAlgebraicElements.AlgebraicNumbers.AlgebraicClosure.

Chapter 30 RulerandCompassConstructiblePointsandNumbers.ImpossibleConstructions.ConstructibleAnglesandPolygons.

Chapter 31 GaloisTheory:PreambleMultipleRoots.RootField.ExtensionofaField.Isomorphism.RootsofUnity.SeparablePolynomials.NormalExtensions.

Chapter 32 GaloisTheory:TheHeartoftheMatterFieldAutomorphisms.TheGaloisGroup.TheGaloisCorrespondence.FundamentalTheoremofGaloisTheory.ComputingGaloisGroups.

Chapter 33 SolvingEquationsbyRadicalsRadicalExtensions.AbelianExtensions.SolvableGroups.InsolvabilityoftheQuintic.

AppendixA ReviewofSetTheory

AppendixB ReviewoftheIntegers

AppendixC ReviewofMathematicalInductionAnswerstoSelectedExercises

Index

*Italicheadingsindicatetopicsdiscussedintheexercisesections.

PREFACE

Once,whenIwasastudentstrugglingtounderstandmodernalgebra,Iwastoldtoviewthissubjectasanintellectualchessgame,withconventionalmovesandprescribedrulesofplay.Iwasillservedbythisbitofextemporaneousadvice,andvowednevertoperpetuatethefalsehoodthatmathematicsispurelyorprimarilyaformalism.Mypledgehasstronglyinfluencedtheshapeandstyleofthisbook.

Whilegivingdueemphasis to thedeductiveaspectofmodernalgebra, Ihaveendeavoredhere topresentmodernalgebraasa livelybranchofmathematics,havingconsiderable imaginativeappealandrestingonsomefirm,clear,andfamiliarintuitions.Ihavedevotedagreatdealofattentiontobringingoutthemeaningfulnessofalgebraicconcepts,bytracingtheseconceptstotheiroriginsinclassicalalgebraandat the same timeexploring their connectionswithotherparts ofmathematics, especiallygeometry,numbertheory,andaspectsofcomputationandequationsolving.

InanintroductorychapterentitledWhyAbstractAlgebra?,aswellasinnumeroushistoricalasides,conceptsof abstract algebra are traced to thehistoric context inwhich theyarose. Ihaveattempted toshowthattheyarosewithoutartifice,asanaturalresponsetoparticularneeds,inthecourseofanaturalprocessofevolution.Furthermore,Ihaveendeavoredtobringtolight,explicitly,theintuitivecontentofthealgebraicconceptsusedinthisbook.Conceptsaremoremeaningfultostudentswhenthestudentsareable to represent those concepts in theirminds by clear and familiarmental images.Accordingly, theprocessofconcreteconcept-formationisdevelopedwithcarethroughoutthisbook.

Ihavedeliberatelyavoidedarigidconventionalformat,withitssuccessionofdefinition, theorem,proof,corollary,example.Inmyexperience,thatkindofformatencouragessomestudentstobelievethatmathematical concepts have a merely conventional character, and may encourage rote memorization.Instead, each chapter has the form of a discussionwith the student,with the accent on explaining andmotivating.

Inanefforttoavoidfragmentationofthesubjectmatterintolooselyrelateddefinitionsandresults,eachchapterisbuiltaroundacentralthemeandremainsanchoredtothisfocalpoint.Inthelaterchaptersespecially,thisfocalpointisaspecificapplicationoruse.Detailsofeverytopicarethenwovenintothegeneraldiscussion,soastokeepanaturalflowofideasrunningthrougheachchapter.

The arrangement of topics is designed to avoid tedious proofs and long-winded explanations.Routine arguments are worked into the discussion whenever this seems natural and appropriate, andproofstotheoremsareseldommorethanafewlineslong.(Thereare,ofcourse,afewexceptionstothis.)Elementarybackgroundmaterial is filled inas it isneeded. For example, a brief chapter on functionsprecedes the discussion of permutation groups, and a chapter on equivalence relations and partitionspavesthewayforLagrangestheorem.

This book addresses itself especially to the average student, to enable him or her to learn andunderstand asmuch algebra as possible. In scope and subject-matter coverage, it is nodifferent frommanyotherstandardtexts.Itbeginswiththepromiseofdemonstratingtheunsolvabilityofthequinticandendswiththatpromisefulfilled.Standardtopicsarediscussedintheirusualorder,andmanyadvanced

and peripheral subjects are introduced in the exercises, accompanied by ample instruction andcommentary.

Ihaveincludedacopioussupplyofexercisesprobablymoreexercisesthaninotherbooksatthislevel.Theyaredesigned tooffer awide rangeof experiences to students atdifferent levelsof ability.Thereissomenoveltyinthewaytheexercisesareorganized:attheendofeachchapter,theexercisesaregroupedintoexercisesets,eachsetcontainingaboutsixtoeightexercisesandheadedbyadescriptivetitle.Eachsettouchesuponanideaorskillcoveredinthechapter.

Thefirstfewexercisesetsineachchaptercontainproblemswhichareessentiallycomputationalormanipulative.Then,therearetwoorthreesetsofsimpleproof-typequestions,whichrequiremainlytheability to put together definitions and results with understanding of their meaning. After that, I haveendeavoredtomaketheexercisesmoreinterestingbyarrangingthemsothatineachsetanewresultisproved,ornewlightisshedonthesubjectofthechapter.

As a rule, all the exercises have the sameweight: very simple exercises are grouped together aspartsofasingleproblem,andconversely,problemswhichrequireacomplexargumentarebrokenintoseveralsubproblemswhichthestudentmaytackleinturn.Ihaveselectedmainlyproblemswhichhaveintrinsic relevance, and are not merely drill, on the premise that this is much more satisfying to thestudent.

CHANGESINTHESECONDEDITIONDuring the seven years that have elapsed since publication of the first edition ofA Book of AbstractAlgebra,Ihavereceivedlettersfrommanyreaderswithcommentsandsuggestions.Moreover,anumberofreviewershavegoneoverthetextwiththeaimoffindingwaystoincreaseitseffectivenessandappealasateachingtool.Inpreparingthesecondedition,Ihavetakenaccountofthemanysuggestionsthatweremade,andofmyownexperiencewiththebookinmyclasses.

Inadditiontonumeroussmallchangesthatshouldmakethebookeasiertoread,thefollowingmajorchangesshouldbenoted:EXERCISESMany of the exercises have been refined or rewordedand a few of the exercise setsreorganizedin order to enhance their clarity or, in some cases, to make them more mathematicallyinteresting.Inaddition,severalnewexericsesetshavebeenincludedwhichtouchuponapplicationsofalgebraandarediscussednext:APPLICATIONSThequestionofincludingapplicationsofabstractalgebrainanundergraduatecourse(especiallyaone-semestercourse)isatouchyone.Eitheronerunstheriskofmakingavisiblyweakcasefor the applicability of the notions of abstract algebra, or on the other handby including substantiveapplicationsonemayenduphavingtoomitalotofimportantalgebra.IhaveadoptedwhatIbelieveisareasonablecompromisebyaddinganelementarydiscussionofafewapplicationareas(chieflyaspectsofcodingandautomatatheory)onlyintheexercisesections,inconnectionwithspecificexercise.Theseexercisesmaybeeitherstressed,de-emphasized,oromittedaltogether.PRELIMINARIESItmaywellbearguedthat,inordertoguaranteethesmootheflowandcontinuityofacourse in abstract algebra, the course should beginwith a review of such preliminaries as set theory,inductionandthepropertiesofintegers.Inordertoprovidematerialforteacherswhoprefertostartthecourseinthisfashion,IhaveaddedanAppendixwiththreebriefchaptersonSets,IntegersandInduction,respectively,eachwithitsownsetofexercises.SOLUTIONSTOSELECTEDEXERCISESAfewexercisesineachchapteraremarkedwiththesymbol#.Thisindicatesthatapartialsolution,orsometimesmerelyadecisivehint,aregivenattheendofthebookinthesectiontitledSolutionstoSelectedExercises.

ACKNOWLEDGMENTSIwouldliketoexpressmythanksforthemanyusefulcommentsandsuggestionsprovidedbycolleagueswhoreviewedthistextduringthecourseofthisrevision,especiallytoJ.RichardByrne,PortlandStateUniversity:D.R.LaTorre,ClemsonUniversity;KazemMahdavi,StateUniversityCollegeatPotsdam;ThomasN.Roe,SouthDakotaStateUniversity;andArmondE.Spencer,StateUniversityofNewYork-Potsdam.Inparticular,IwouldliketothankRobertWeinstein,mathematicseditoratMcGraw-Hillduringthe preparation of the second edition of this book. I am indebted to him for his guidance, insight, andsteadyencouragement.

CharlesC.Pinter

CHAPTER

ONEWHYABSTRACTALGEBRA?

Whenweopenatextbookofabstractalgebraforthefirst timeandperusethetableofcontents,wearestruckbytheunfamiliarityofalmosteverytopicweseelisted.Algebraisasubjectweknowwell,buthereitlookssurprisinglydifferent.Whatarethesedifferences,andhowfundamentalarethey?

First,thereisamajordifferenceinemphasis.Inelementaryalgebrawelearnedthebasicsymbolismandmethodologyofalgebra;wecametoseehowproblemsoftherealworldcanbereducedtosetsofequations and how these equations can be solved to yield numerical answers. This technique fortranslatingcomplicatedproblems into symbols is thebasis for all furtherwork inmathematics and theexactsciences,andisoneofthetriumphsofthehumanmind.However,algebraisnotonlyatechnique,itisalsoabranchoflearning,adiscipline,likecalculusorphysicsorchemistry.Itisacoherentandunifiedbodyofknowledgewhichmaybestudiedsystematically,startingfromfirstprinciplesandbuildingup.Sothe first difference between the elementary and themore advanced course in algebra is that, whereasearlierweconcentratedontechnique,wewillnowdevelopthatbranchofmathematicscalledalgebrainasystematicway.Ideasandgeneralprincipleswilltakeprecedenceoverproblemsolving.(Bytheway,thisdoesnotmean thatmodernalgebrahasnoapplicationsquite theopposite is true,aswewillseesoon.)

Algebraatthemoreadvancedlevelisoftendescribedasmodernorabstractalgebra.Infact,bothofthesedescriptionsarepartlymisleading.Someofthegreatdiscoveriesintheupperreachesofpresent-dayalgebra(forexample,theso-calledGaloistheory)wereknownmanyyearsbeforetheAmericanCivilWar;andthebroadaimsofalgebratodaywereclearlystatedbyLeibnizintheseventeenthcentury.Thus,modernalgebraisnotsoverymodern,afterall!Towhatextentisitabstract?Well,abstractionisallrelative; one persons abstraction is another persons bread and butter. The abstract tendency inmathematicsisalittlelikethesituationofchangingmoralcodes,orchangingtastesinmusic:Whatshocksonegenerationbecomesthenorminthenext.Thishasbeentruethroughoutthehistoryofmathematics.

Forexample,1000yearsagonegativenumberswereconsideredtobeanoutrageousidea.Afterall,itwassaid,numbersareforcounting:wemayhaveoneorange,ortwooranges,ornoorangesatall;buthow canwe haveminus an orange? The logisticians, or professional calculators, of those days usednegativenumbersasanaidintheircomputations;theyconsideredthesenumberstobeausefulfiction,forifyoubelieveinthemtheneverylinearequationax+b=0hasasolution(namelyx=b/a,provideda0).EventhegreatDiophantusoncedescribedthesolutionof4x+6=2asanabsurdnumber.Theideaofa systemof numerationwhich includednegativenumberswas far too abstract formanyof the learnedheadsofthetenthcentury!

The history of the complex numbers (numberswhich involve ) is verymuch the same. Forhundredsofyears,mathematiciansrefusedtoacceptthembecausetheycouldntfindconcreteexamplesorapplications.(Theyarenowabasictoolofphysics.)

Settheorywasconsideredtobehighlyabstractafewyearsago,andsowereothercommonplacesoftoday.Manyof theabstractionsofmodernalgebraarealreadybeingusedby scientists, engineers, andcomputerspecialistsintheireverydaywork.Theywillsoonbecommonfare,respectablyconcrete,andbythentherewillbenewabstractions.

Laterinthischapterwewilltakeacloserlookattheparticularbrandofabstractionusedinalgebra.Wewillconsiderhowitcameaboutandwhyitisuseful.

Algebrahasevolvedconsiderably,especiallyduringthepast100years.Itsgrowthhasbeencloselylinked with the development of other branches of mathematics, and it has been deeply influenced byphilosophicalideasonthenatureofmathematicsandtheroleoflogic.Tohelpusunderstandthenatureandspiritofmodernalgebra,weshouldtakeabrieflookatitsorigins.

ORIGINSTheorderinwhichsubjectsfolloweachotherinourmathematicaleducationtendstorepeatthehistoricalstages in the evolution of mathematics. In this scheme, elementary algebra corresponds to the greatclassicalageofalgebra,whichspansabout300yearsfromthesixteenththroughtheeighteenthcenturies.It was during these years that the art of solving equations became highly developed and modernsymbolismwasinvented.

Theword algebraal jebr in Arabicwas first used byMohammed of Kharizm, who taughtmathematicsinBaghdadduringtheninthcentury.Thewordmayberoughlytranslatedasreunion,anddescribeshismethodforcollectingthetermsofanequationinordertosolveit.Itisanamusingfactthatthe word algebra was first used in Europe in quite another context. In Spain barbers were calledalgebristas,orbonesetters(theyreunitedbrokenbones),becausemedievalbarbersdidbonesettingandbloodlettingasasidelinetotheirusualbusiness.

Theoriginof thewordclearlyreflects theactualcontextofalgebraat that time,for itwasmainlyconcerned with ways of solving equations. In fact, Omar Khayyam, who is best remembered for hisbrilliantversesonwine,song, love,andfriendshipwhicharecollectedintheRubaiyatbutwhowasalsoagreatmathematicianexplicitlydefinedalgebraasthescienceofsolvingequations.

Thus, aswe enter upon the threshold of the classical age of algebra, its central theme is clearlyidentifiedasthatofsolvingequations.Methodsofsolvingthelinearequationax+b=0andthequadraticax2+bx+c=0werewellknownevenbeforetheGreeks.Butnobodyhadyetfoundageneralsolutionforcubicequations

x3+ax2+bx=c

orquartic(fourth-degree)equations

x4+ax3+bx2+cx=d

Thisgreataccomplishmentwasthetriumphofsixteenthcenturyalgebra.The setting is Italy and the time is the Renaissancean age of high adventure and brilliant

achievement,whenthewideworldwasreawakeningafterthelongausterityoftheMiddleAges.Americahadjustbeendiscovered,classicalknowledgehadbeenbroughttolight,andprosperityhadreturnedto

thegreatcitiesofEurope.Itwasaheadyagewhennothingseemedimpossibleandeventheoldbarriersof birth and rank could be overcome. Courageous individuals set out for great adventures in the farcornersoftheearth,whileothers,nowconfidentonceagainofthepowerofthehumanmind,wereboldlyexploringthelimitsofknowledgeinthesciencesandthearts.Theidealwastobeboldandmany-faceted,toknowsomethingofeverything,andeverythingofatleastonething.Thegreattraderswerepatronsofthearts,thefinestmindsinsciencewereadeptsatpoliticalintrigueandhighfinance.Thestudyofalgebrawasreborninthislivelymilieu.

Thosemenwhobroughtalgebratoahighlevelofperfectionatthebeginningofitsclassicalagealltypical products of the Italian Renaissaneewere as colorful and extraordinary a lot as have everappeared in a chapter of history. Arrogant and unscrupulous, brilliant, flamboyant, swaggering, andremarkable,theylivedtheirlivesastheydidtheirwork:withstyleandpanache,inbrilliantdashesandinspiredleapsoftheimagination.

The spirit of scholarship was not exactly as it is today. These men, instead of publishing theirdiscoveries, kept them as well-guarded secrets to be used against each other in problem-solvingcompetitions.Such contestswere a popular attraction: heavybetsweremadeon the rival parties, andtheirreputations(aswellasasubstantialpurse)dependedontheoutcome.

Oneof themost remarkableof thesemenwasGirolamoCardan.Cardanwasborn in1501as theillegitimatesonofafamousjuristofthecityofPavia.Amanofpassionatecontrasts,hewasdestinedtobecomefamousasaphysician,astrologer,andmathematicianandnotoriousasacompulsivegambler,scoundrel,andheretic.Afterhegraduatedinmedicine,hiseffortstobuildupamedicalpracticeweresounsuccessfulthatheandhiswifewereforcedtoseekrefugeinthepoorhouse.Withthehelpoffriendshebecame a lecturer inmathematics, and, after he cured the child of a senator fromMilan, hismedicalcareeralsopickedup.Hewasfinallyadmittedtothecollegeofphysiciansandsoonbecameitsrector.Abrilliantdoctor,hegavethefirstclinicaldescriptionoftyphusfever,andashisfamespreadhebecamethepersonalphysicianofmanyofthehighandmightyofhisday.

Cardansearlyinterestinmathematicswasnotwithoutapracticalside.Asaninveterategamblerhewas fascinatedbywhathe recognized tobe the lawsofchance.HewroteagamblersmanualentitledBookonGamesofChance,whichpresents the first systematic computationsof probabilities.He alsoneededmathematics as a tool in casting horoscopes, for his fame as an astrologer was great and hispredictionswerehighlyregardedandsoughtafter.Hismostimportantachievementwasthepublicationofa book called Ars Magna (The Great Art), in which he presented systematically all the algebraicknowledgeofhistime.However,asalreadystated,muchofthisknowledgewasthepersonalsecretofitspractitioners, and had to be wheedled out of them by cunning and deceit. The most importantaccomplishment of the day, the general solution of the cubic equation which had been discovered byTartaglia,wasobtainedinthatfashion.

Tartagliaslifewasasturbulentasanyinthosedays.BornwiththenameofNiccoloFontanaabout1500,hewaspresentattheoccupationofBresciabytheFrenchin1512.Heandhisfatherfledwithmanyothersintoacathedralforsanctuary,butintheheatofbattlethesoldiersmassacredthehaplesscitizenseveninthatholyplace.Thefatherwaskilled,andtheboy,withasplitskullandadeepsabercutacrosshisjawsandpalate,wasleftfordead.Atnighthismotherstoleintothecathedralandmanagedtocarryhimoff;miraculouslyhesurvived.Thehorrorofwhathehadwitnessedcausedhimtostammerfor therestofhislife,earninghimthenicknameTartaglia,thestammerer,whichheeventuallyadopted.

Tartaglia receivedno formal schooling, for thatwasaprivilegeof rankandwealth.However,hetaughthimselfmathematicsandbecameoneof themostgiftedmathematiciansofhisday.He translatedEuclid and Archimedes and may be said to have originated the science of ballistics, for he wrote atreatiseongunnerywhichwasapioneeringeffortonthelawsoffallingbodies.

In1535Tartagliafoundawayofsolvinganycubicequationoftheformx3+ax2=b(thatis,withoutan x term). When be announced his accomplishment (without giving any details, of course), he waschallenged to an algebra contest by a certain Antonio Fior, a pupil of the celebrated professor ofmathematicsScipiodelFerro.Scipiohadalreadyfoundamethodforsolvinganycubicequationoftheformx3+ax=b(thatis,withoutanx2term),andhadconfidedhissecrettohispupilFior.Itwasagreedthateachcontestantwastodrawup30problemsandhandthelisttohisopponent.Whoeversolvedthegreaternumberofproblemswouldreceiveasumofmoneydepositedwithalawyer.Afewdaysbeforethecontest,Tartagliafoundawayofextendinghismethodsoastosolveanycubicequation.Inlessthan2hours he solved all his opponents problems, while his opponent failed to solve even one of thoseproposedbyTartaglia.

For some timeTartagliakepthismethod for solvingcubicequations tohimself,but in theendhesuccumbedtoCardansaccomplishedpowersofpersuasion.InfluencedbyCardanspromisetohelphimbecomeartilleryadvisertotheSpanisharmy,herevealedthedetailsofhismethodtoCardanunderthepromiseofstrictsecrecy.Afewyearslater,toTartagliasunbelievingamazementandindignation,CardanpublishedTartagliasmethod in his bookArsMagna. Even though he gaveTartaglia full credit as theoriginatorofthemethod,therecanbenodoubtthathebrokehissolemnpromise.Abitterdisputearosebetweenthemathematicians,fromwhichTartagliawasperhapsluckytoescapealive.HelosthispositionaspubliclectureratBrescia,andlivedouthisremainingyearsinobscurity.

Thenextgreatstepintheprogressofalgebrawasmadebyanothermemberofthesamecircle.ItwasLudovico Ferrari who discovered the general method for solving quartic equationsequations of theform

x4+ax3+bx2+cx=d

Ferrari was Cardans personal servant. As a boy in Cardans service he learned Latin, Greek, andmathematics.HewonfameafterdefeatingTartagliainacontestin1548,andreceivedanappointmentassupervisoroftaxassessmentsinMantua.Thispositionbroughthimwealthandinfluence,buthewasnotabletodominatehisownviolentdisposition.HequarreledwiththeregentofMantua,losthisposition,anddiedattheageof43.Traditionhasitthathewaspoisonedbyhissister.

As for Cardan, after a long career of brilliant and unscrupulous achievement, his luck finallyabandonedhim.Cardans son poisoned his unfaithfulwife andwas executed in 1560.Ten years later,Cardan was arrested for heresy because he published a horoscope of Christs life. He spent severalmonthsinjailandwasreleasedafterrenouncinghisheresyprivately,butlosthisuniversitypositionandthe right topublishbooks.Hewas leftwitha smallpensionwhichhadbeengranted tohim, for someunaccountablereason,bythePope.

Asthiscolorfultimedrawstoaclose,algebraemergesasamajorbranchofmathematics.Itbecameclear thatmethodscanbe found to solvemanydifferent typesof equations. Inparticular, formulashadbeen discovered which yielded the roots of all cubic and quartic equations. Now the challenge wasclearlyouttotakethenextstep,namely,tofindaformulafortherootsofequationsofdegree5orhigher(inotherwords,equationswithanx5term,oranx6term,orhigher).Duringthenext200years,therewashardlyamathematicianofdistinctionwhodidnottrytosolvethisproblem,butnonesucceeded.Progresswasmade in newparts of algebra, and algebrawas linked to geometrywith the invention of analyticgeometry.Buttheproblemofsolvingequationsofdegreehigherthan4remainedunsettled.Itwas,intheexpressionofLagrange,achallengetothehumanmind.

Itwasthereforeagreatsurprisetoallmathematicianswhenin1824theworkofayoungNorwegianprodigynamedNielsAbelcametolight.Inhiswork,Abelshowedthattheredoesnotexistanyformula

(intheconventionalsensewehaveinmind)fortherootsofanalgebraicequationwhosedegreeis5orgreater. This sensational discovery brings to a close what is called the classical age of algebra.Throughoutthisagealgebrawasconceivedessentiallyasthescienceofsolvingequations,andnowtheouterlimitsofthisquesthadapparentlybeenreached.Intheyearsahead,algebrawastostrikeoutinnewdirections.

THEMODERNAGEAbout the time Niels Abel made his remarkable discovery, several mathematicians, workingindependentlyindifferentpartsofEurope,beganraisingquestionsaboutalgebrawhichhadneverbeenconsidered before. Their researches in different branches of mathematics had led them to investigatealgebrasofaveryunconventionalkindandinconnectionwiththesealgebrastheyhadtofindanswerstoquestionswhichhadnothingtodowithsolvingequations.Theirworkhadimportantapplications,andwassoontocompelmathematicianstogreatlyenlargetheirconceptionofwhatalgebraisabout.

The new varieties of algebra arose as a perfectly natural development in connection with theapplicationofmathematicstopracticalproblems.Thisiscertainlytruefortheexampleweareabouttolookatfirst.

TheAlgebraofMatricesAmatrixisarectangulararrayofnumberssuchas

Such arrays come up naturally inmany situations, for example, in the solution of simultaneous linearequations.Theabovematrix,forinstance,isthematrixofcoefficientsofthepairofequations

Sincethesolutionofthispairofequationsdependsonlyonthecoefficients,wemaysolveitbyworkingonthematrixofcoefficientsaloneandignoringeverythingelse.

Wemayconsidertheentriesofamatrixtobearrangedinrowsandcolumns;theabovematrixhastworowswhichare

(2113)and(90.54)

andthreecolumnswhichare

Itisa23matrix.Tosimplifyourdiscussion,wewillconsideronly22matricesintheremainderofthissection.Matricesareaddedbyaddingcorrespondingentries:

Thematrix

iscalledthezeromatrixandbehaves,underaddition,likethenumberzero.Themultiplicationofmatricesisalittlemoredifficult.First,letusrecallthatthedotproductoftwo

vectors(a,b)and(a,b)is

(a,b)(a,b)=aa+bb

thatis,wemultiplycorrespondingcomponentsandadd.Now,supposewewanttomultiplytwomatricesAandB;weobtaintheproductABasfollows:

TheentryinthefirstrowandfirstcolumnofAB,thatis,inthisposition

isequaltothedotproductofthefirstrowofAbythefirstcolumnofB.TheentryinthefirstrowandsecondcolumnofAB,inotherwords,thisposition

isequaltothedotproductofthefirstrowofAbythesecondcolumnofB.Andsoon.Forexample,

Sofinally,

The rulesof algebra formatricesareverydifferent from the rulesof conventionalalgebra.Forinstance,thecommutativelawofmultplica-tion,AB=BA,isnottrue.Hereisasimpleexample:

IfAisarealnumberandA2=0,thennecessarilyA=0;butthisisnottrueofmatrices.Forexample,

thatis,A2=0althoughA0.Inthealgebraofnumbers,ifAB=ACwhereA0,wemaycancelAandconcludethatB=C. In

matrixalgebrawecannot.Forexample,

thatis,AB=AC,A0,yetBC.Theidentitymatrix

correspondsinmatrixmultiplicationtothenumber1;forwehaveAI=IA=Aforevery22matrixA.IfAisanumberandA2=1,weconcludethatA=1Matricesdonotobeythisrule.Forexample,

thatis,A2=I,andyetAisneitherInorI.Nomorewillbesaidaboutthealgebraofmatricesatthispoint,exceptthatwemustbeaware,once

again,thatitisanewgamewhoserulesarequitedifferentfromthoseweapplyinconventionalalgebra.

BooleanAlgebraAn evenmore bizarre kind of algebrawas developed in themid-nineteenth century by anEnglishmannamed George Boole. This algebrasubsequently named boolean algebra after its inventorhas amyriadofapplicationstoday.Itisformallythesameasthealgebraofsets.

IfS isaset,wemayconsiderunionand intersection tobeoperationson thesubsetsof5.Letusagreeprovisionallytowrite

A+B for AB

and

AB for AB

(Thisconventionisnotunusual.)Then,

andsoon.Theseidentitiesareanalogoustotheonesweuseinelementaryalgebra.Butthefollowingidentities

arealsotrue,andtheyhavenocounterpartinconventionalalgebra:

andsoon.This unusual algebra has become a familiar tool for peoplewhoworkwith electrical networks,

computersystems,codes,andsoon.Itisasdifferentfromthealgebraofnumbersasitisfromthealgebraofmatrices.

Other exotic algebras arose in avarietyof contexts, often in connectionwith scientific problems.Therewerecomplexandhypercomplexalgebras,algebrasofvectorsandtensors,andmanyothers.Todayitisestimatedthatover200differentkindsofalgebraicsystemshavebeenstudied,eachofwhicharoseinconnectionwithsomeapplicationorspecificneed.

AlgebraicStructuresAs legions of new algebras began to occupy the attention ofmathematicians, the awareness grew thatalgebracannolongerbeconceivedmerelyasthescienceofsolvingequations.Ithadtobeviewedmuchmorebroadlyasabranchofmathematicscapableofrevealinggeneralprincipleswhichapplyequallytoallknownandallpossiblealgebras.

What is it thatallalgebrashaveincommon?What traitdotheysharewhichletsusrefer toallofthemasalgebras?Inthemostgeneralsense,everyalgebraconsistsofaset(asetofnumbers,asetofmatrices,asetofswitchingcomponents,oranyotherkindofset)andcertainoperationsonthatset.Anoperationissimplyawayofcombininganytwomembersofasettoproduceauniquethirdmemberofthesameset.

Thus,weareledtothemodernnotionofalgebraicstructure.Analgebraicstructureisunderstoodtobeanarbitrary set,withoneormoreoperationsdefinedon it.Andalgebra, then, isdefined tobe thestudyofalgebraicstructures.

Itisimportantthatwebeawakenedtothefullgeneralityofthenotionofalgebraicstructure.Wemustmakeanefforttodiscardallourpreconceivednotionsofwhatanalgebrais,andlookatthisnewnotionofalgebraicstructureinitsnakedsimplicity.Anyset,witharule(orrules)forcombiningitselements,isalreadyanalgebraicstructure.Theredoesnotneed tobeanyconnectionwithknownmathematics.Forexample,considerthesetofallcolors(purecolorsaswellascolorcombinations),andtheoperationofmixinganytwocolorstoproduceanewcolor.Thismaybeconceivedasanalgebraicstructure.Itobeyscertainrules,suchasthecommutativelaw(mixingredandblueisthesameasmixingblueandred).Inasimilarvein,consider thesetofallmusicalsoundswith theoperationofcombiningany twosounds toproduceanew(harmoniousordisharmonious)combination.

Asanotherexample,imaginethattheguestsatafamilyreunionhavemadeuparuleforpickingthe

closestcommonrelativeofanytwopersonspresentatthereunion(andsupposethat,foranytwopeopleat the reunion, their closest common relative is also present at the reunion). This too, is an algebraicstructure:wehaveaset(namelythesetofpersonsatthereunion)andanoperationonthatset(namelytheclosestcommonrelativeoperation).

Asthegeneralnotionofalgebraicstructurebecamemorefamiliar(itwasnotfullyaccepteduntiltheearlypartof the twentiethcentury), itwasbound tohaveaprofound influenceonwhatmathematiciansperceived algebra tobe. In the end it became clear that the purpose of algebra is to study algebraicstructures,andnothinglessthanthat.Ideallyitshouldaimtobeageneralscienceofalgebraicstructureswhoseresultsshouldhaveapplicationstoparticularcases,therebymakingcontactwiththeolderpartsofalgebra.Beforewetakeacloserlookatthisprogram,wemustbrieflyexamineanotheraspectofmodernmathematics,namely,theincreasinguseoftheaxiomaticmethod.

AXIOMSTheaxiomaticmethodisbeyonddoubtthemostremarkableinventionofantiquity,andinasensethemostpuzzling. It appeared suddenly inGreek geometry in a highly developed formalready sophisticated,elegant,andthoroughlymoderninstyle.NothingseemstohaveforeshadoweditanditwasunknowntoancientmathematiciansbeforetheGreeks.Itappearsforthefirsttimeinthelightofhistoryinthegreattextbook of early geometry, Euclids Elements. Its originsthe first tentative experiments in formaldeductivereasoningwhichmusthaveprecededitremainsteepedinmystery.

EuclidsElements embodies the axiomaticmethod in its purest form.This amazingbookcontains465 geometric propositions, some fairly simple, some of astounding complexity. What is reallyremarkable,though,isthatthe465propositions,formingthelargestbodyofscientificknowledgeintheancientworld,arederivedlogicallyfromonly10premiseswhichwouldpassastrivialobservationsofcommonsense.Typicalofthepremisesarethefollowing:

Thingsequaltothesamethingareequaltoeachother.Thewholeisgreaterthanthepart.Astraightlinecanbedrawnthroughanytwopoints.Allrightanglesareequal.

So great was the impressionmade by EuclidsElements on following generations that it became themodelofcorrectmathematicalformandremainssotothisday.

ItwouldbewrongtobelievetherewasnonotionofdemonstrativemathematicsbeforethetimeofEuclid. There is evidence that the earliest geometers of the ancient Middle East used reasoning todiscovergeometricprinciples.Theyfoundproofsandmusthavehituponmanyofthesameproofswefindin Euclid. The difference is that Egyptian and Babylonian mathematicians considered logicaldemonstration tobeanauxiliaryprocess, like thepreliminarysketchmadebyartistsaprivatementalprocesswhichguidedthemtoaresultbutdidnotdeservetoberecorded.Suchanattitudeshowslittleunderstandingofthetruenatureofgeometryanddoesnotcontaintheseedsoftheaxiomaticmethod.

It is also known today thatmanymaybemostof the geometric theorems in EuclidsElementscame frommoreancient times, andwereprobablyborrowedbyEuclid fromEgyptianandBabyloniansources.However,thisdoesnotdetractfromthegreatnessofhiswork.ImportantasarethecontentsoftheElements,what has proved farmore important for posterity is the formalmanner inwhichEuclidpresented these contents.Theheart of thematterwas thewayheorganized geometric factsarrangedthemintoalogicalsequencewhereeachtheorembuildsonprecedingtheoremsandthenformsthelogical

basisforothertheorems.(Wemustcarefullynotethattheaxiomaticmethodisnotawayofdiscoveringfactsbutoforganizing

them.Newfactsinmathematicsarefound,asoftenasnot,byinspiredguessesorexperiencedintuition.Tobeaccepted,however,theyshouldbesupportedbyproofinanaxiomaticsystem.)

EuclidsElementshasstoodthroughouttheagesasthemodeloforganized,rationalthoughtcarriedto itsultimateperfection.Mathematiciansandphilosophers ineverygenerationhave tried to imitate itslucid perfection and flawless simplicity. Descartes and Leibniz dreamed of organizing all humanknowledge into an axiomatic system, andSpinoza created a deductive systemof ethics patterned afterEuclidsgeometry.Whilemanyofthesedreamshaveprovedtobeimpractical,themethodpopularizedbyEuclidhasbecometheprototypeofmodernmathematicalform.Sincethemiddleofthenineteenthcentury,theaxiomaticmethodhasbeenacceptedastheonlycorrectwayoforganizingmathematicalknowledge.

Toperceivewhy theaxiomaticmethod is trulycentral tomathematics,wemustkeepone thing inmind:mathematicsby itsnature isessentiallyabstract.Forexample, ingeometry straight linesarenotstretchedthreads,butaconceptobtainedbydisregardingallthepropertiesofstretchedthreadsexceptthatofextendinginonedirection.Similarly,theconceptofageometricfigureistheresultofidealizingfromallthepropertiesofactualobjectsandretainingonlytheirspatialrelationships.Now,sincetheobjectsofmathematicsareabstractions,itstandstoreasonthatwemustacquireknowledgeaboutthembylogicandnotbyobservationorexperiment(forhowcanoneexperimentwithanabstractthought?).

Thisremarkappliesveryaptlytomodernalgebra.Thenotionofalgebraicstructureisobtainedbyidealizing from all particular, concrete systems of algebra.We choose to ignore the properties of theactualobjectsinasystemofalgebra(theymaybenumbers,ormatrices,orwhateverwedisregardwhattheyare),andweturnourattentionsimplytothewaytheycombineunderthegivenoperations.Infact,justaswedisregardwhattheobjectsinasystemare,wealsodisregardwhattheoperationsdotothem.We retain only the equations and inequalitieswhich hold in the system, for only these are relevant toalgebra.Everythingelsemaybediscarded.Finally,equationsandinequalitiesmaybededucedfromoneanotherlogically,justasspatialrelationshipsarededucedfromeachotheringeometry.

THEAXIOMATICSOFALGEBRALetusrememberthat in themid-nineteenthcentury,wheneccentricnewalgebrasseemedtoshowupateveryturninmathematicalresearch,itwasfinallyunderstoodthatsacrosanctlawssuchastheidentitiesab =ba and a(bc) = (ab)c are not inviolablefor there are algebras in which they do not hold. Byvaryingordeletingsomeof these identities,orbyreplacingthembynewones,anenormousvarietyofnewsystemscanbecreated.

Most importantly,mathematiciansslowlydiscovered thatall thealgebraic lawswhichhold inanysystemcanbederivedfromafewsimple,basicones.Thisisagenuinelyremarkablefact,foritparallelsthediscoverymadebyEuclidthatafewverysimplegeometricpostulatesaresufficienttoproveallthetheorems of geometry. As it turns out, then, we have the same phenomenon in algebra: a few simplealgebraicequationsofferthemselvesnaturallyasaxioms,andfromthemallotherfactsmaybeproved.

Thesebasicalgebraic lawsare familiar tomosthigh school students today.We list themhere forreference.WeassumethatAisanysetandthereisanoperationonAwhichwedesignatewiththesymbol*

a*b=b*a (1)IfEquation(1)istrueforanytwoelementsaandbinA,wesaythattheoperation*iscommutative.Whatitmeans,ofcourse,isthatthevalueofa*b(orb*a)isindependentoftheorderinwhichaandbaretaken.

a*(b*c)=(a*b)*c (2)IfEquation(2) is true for any three elementsa,b, and c inA, we say the operation * isassociative.Remember thatanoperation isa rule forcombiningany twoelements,so ifwewant tocombine threeelements,we can do so in differentways. Ifwewant to combinea,b, and cwithout changing theirorder,wemayeithercombineawiththeresultofcombiningbandc,whichproducesa*(b*c);orwemayfirstcombineawithb,andthencombinetheresultwithc,producing(a*b)*c.Theassociativelawassertsthatthesetwopossiblewaysofcombiningthreeelements(withoutchangingtheirorder)yieldthesameresult.

ThereexistsanelementeinAsuchthate*a=a and a*e=a foreveryainA (3)

IfsuchanelementeexistsinA,wecallitanidentityelementfortheoperation*.Anidentityelementissometimescalledaneutralelement,foritmaybecombinedwithanyelementawithoutalteringa.Forexample,0isanidentityelementforaddition,and1isanidentityelementformultiplication.

ForeveryelementainA,thereisanelemental(ainverse)inAsuchthata*al=e and a1*a=e (4)

Ifstatement(4)istrueinasystemofalgebra,wesaythateveryelementhasaninversewithrespecttotheoperation*.Themeaningoftheinverseshouldbeclear:thecombinationofanyelementwithitsinverseproducestheneutralelement(onemightroughlysaythattheinverseofaneutralizesa).Forexample,ifA is a set of numbers and the operation is addition, then the inverse of any number a is (a); if theoperationismultiplication,theinverseofanya0is1/a.

Let us assumenow that the same setA has a secondoperation, symbolized by, aswell as theoperation*:

a*(bc)=(a*b)(a*c) (5)IfEquation(5)holdsforanythreeelementsa,b,andcinA,wesaythat*isdistributiveover.Ifthereare two operations in a system, theymust interact in someway; otherwise therewould be no need toconsiderthemtogether.Thedistributivelawisthemostcommonway(butnottheonlypossibleone)fortwooperationstoberelatedtooneanother.

There are other basic laws besides the fivewehave just seen, but these are themost commonones.Themostimportantalgebraicsystemshaveaxiomschosenfromamongthem.Forexample,whenamathematiciannowadaysspeaksofaring,themathematicianisreferringtoasetAwithtwooperations,usuallysymbolizedby+and,havingthefollowingaxioms:

Additioniscommutativeandassociative,ithasaneutralelementcommonlysymbolizedby0,andeveryelementahasaninverseawithrespecttoaddition.Multiplicationisassociative,hasaneutralelement1,andisdistributiveoveraddition.

Matrixalgebraisaparticularexampleofaring,andallthelawsofmatrixalgebramaybeprovedfromthe preceding axioms. However, there are many other examples of rings: rings of numbers, rings offunctions,ringsofcodewords,ringsofswitchingcomponents,andagreatmanymore.Everyalgebraiclawwhichcanbeprovedinaring(fromtheprecedingaxioms)istrueineveryexampleofaring.Inotherwords, insteadofproving thesameformularepeatedlyoncefornumbers,onceformatrices,onceforswitchingcomponents,andsoonitissufficientnowadaystoproveonlythattheformulaholdsinrings,andthenofnecessityitwillbetrueinallthehundredsofdifferentconcreteexamplesofrings.

Byvaryingthepossiblechoicesofaxioms,wecankeepcreatingnewaxiomaticsystemsofalgebraendlessly.Wemaywellask: is it legitimatetostudyanyaxiomaticsystem,withanychoiceofaxioms,regardlessofusefulness,relevance,orapplicability?Thereareradicalsinmathematicswhoclaimthe

freedomformathematicians to studyanysystem theywish,without theneed to justify it.However, thepracticeinestablishedmathematics ismoreconservative:particularaxiomaticsystemsareinvestigatedonaccountoftheirrelevancetonewandtraditionalproblemsandotherpartsofmathematics,orbecausetheycorrespondtoparticularapplications.

Inpractice,howisaparticularchoiceofalgebraicaxiomsmade?Verysimply:whenmathematicianslook at different parts of algebra and notice that a common pattern of proofs keeps recurring, andessentiallythesameassumptionsneedtobemadeeachtime,theyfinditnaturaltosingleoutthischoiceofassumptionsastheaxiomsforanewsystem.All theimportantnewsystemsofalgebrawerecreatedinthisfashion.

ABSTRACTIONREVISITEDAnother important aspect of axiomaticmathematics is this:whenwe capturemathematical facts in anaxiomaticsystem,wenevertrytoreproducethefactsinfull,butonlythatsideofthemwhichisimportantorrelevantinaparticularcontext.Thisprocessofselectingwhatisrelevantanddisregardingeverythingelseistheveryessenceofabstraction.

Thiskindofabstractionissonaturaltousashumanbeingsthatwepracticeitallthetimewithoutbeingawareofdoingso.LiketheBourgeoisGentlemaninMoliresplaywhowasamazedtolearnthathespokeinprose,someofusmaybesurprisedtodiscoverhowmuchwethinkinabstractions.Naturepresentsuswithamyriadofinterwovenfactsandsensations,andwearechallengedateveryinstanttosingleoutthosewhichareimmediatelyrelevantanddiscardtherest.Inordertomakeoursurroundingscomprehensible,wemustcontinuallypickoutcertaindataandseparatethemfromeverythingelse.

Fornaturalscientists,thisprocessistheverycoreandessenceofwhattheydo.Natureisnotmadeup of forces, velocities, and moments of inertia. Nature is a wholenature simply is! The physicistisolatescertainaspectsofnaturefromtherestandfindsthelawswhichgoverntheseabstractions.

Itisthesamewithmathematics.Forexample,thesystemoftheintegers(wholenumbers),asknownbyourintuition,isacomplexrealitywithmanyfacets.Themathematicianseparatesthesefacetsfromoneanotherandstudiesthemindividually.Fromonepointofviewthesetoftheintegers,withadditionandmultiplication,formsaring(thatis,itsatisfiestheaxiomsstatedpreviously).Fromanotherpointofviewit isanorderedset,andsatisfiesspecialaxiomsofordering.Onadifferent level, thepositive integersform the basis of recursion theory, which singles out the particular way positive integers may beconstructed,beginningwith1andadding1eachtime.

It therefore happens that the traditional subdivision ofmathematics into subjectmatters has beenradicallyaltered.Nolongeraretheintegersonesubject,complexnumbersanother,matricesanother,andso on; instead, particular aspects of these systems are isolated, put in axiomatic form, and studiedabstractly without reference to any specific objects. The other side of the coin is that each aspect issharedbymanyofthetraditionalsystems:forexample,algebraicallytheintegersformaring,andsodothecomplexnumbers,matrices,andmanyotherkindsofobjects.

Thereisnothingintrinsicallynewaboutthisprocessofdivorcingpropertiesfromtheactualobjectshavingtheproperties;aswehaveseen,itispreciselywhatgeometryhasdoneformorethan2000years.Somehow,ittooklongerforthisprocesstotakeholdinalgebra.

Themovementtowardaxiomaticsandabstractioninmodernalgebrabeganaboutthe1830sandwascompleted 100years later.Themovementwas tentative at first, not quite conscious of its aims, but itgainedmomentumasitconvergedwithsimilartrendsinotherpartsofmathematics.Thethinkingofmanygreatmathematiciansplayedadecisive role,butnone leftadeeperor longer lasting impression thanaveryyoungFrenchmanbythenameofvaristeGalois.

ThestoryofvaristeGaloisisprobablythemostfantasticandtragicinthehistoryofmathematics.Asensitiveandprodigiouslygiftedyoungman,hewaskilledinaduelattheageof20,endingalifewhichinitsbriefspanhadofferedhimnothingbuttragedyandfrustration.Whenhewasonlyayouthhisfathercommitedsuicide,andGaloiswaslefttofendforhimselfinthelabyrinthineworldofFrenchuniversitylife and student politics. He was twice refused admittance to the Ecole Polytechnique, the mostprestigiousscientificestablishmentofitsday,probablybecausehisanswerstotheentranceexaminationwere toooriginal andunorthodox.Whenhepresented an earlyversionofhis importantdiscoveries inalgebratothegreatacademicianCauchy,thisgentlemandidnotreadtheyoungstudentspaper,butlostit.Later,GaloisgavehisresultstoFourierinthehopeofwinningthemathematicsprizeoftheAcademyofSciences. But Fourier died, and that paper, too, was lost. Another paper submitted to Poisson waseventuallyreturnedbecausePoissondidnothavetheinteresttoreaditthrough.

Galois finally gained admittance to the cole Normale, another focal point of research inmathematics,buthewassoonexpelledforwritinganessaywhichattackedtheking.Hewasjailedtwicefor political agitation in the studentworld of Paris. In themidst of such a turbulent life, it is hard tobelievethatGaloisfoundtimetocreatehiscolossallyoriginaltheoriesonalgebra.

WhatGaloisdidwastotieintheproblemoffindingtherootsofequationswithnewdiscoveriesongroupsofpermutations.Heexplainedexactlywhichequationsofdegree5orhigherhavesolutionsofthetraditional kindandwhich others do not.Along theway, he introduced some amazingly original andpowerfulconcepts,whichformtheframeworkofmuchalgebraicthinkingtothisday.AlthoughGaloisdidnotworkexplicitlyinaxiomaticalgebra(whichwasunknowninhisday),theabstractnotionofalgebraicstructureisclearlyprefiguredinhiswork.

In1832,whenGaloiswasonly20yearsold,hewaschallengedtoaduel.Whatargumentledtothechallengeisnotclear:somesaytheissuewaspolitical,whileothersmaintaintheduelwasfoughtoverafickle ladyswavering love. The truthmay never be known, but the turbulent, brilliant, and idealisticGaloisdiedofhiswounds.Fortunatelyformathematics,thenightbeforetheduelhewrotedownhismainmathematical results and entrusted them to a friend. This time, theywerent lostbut theywere onlypublished15yearsafterhisdeath.Themathematicalworldwasnotreadyforthembeforethen!

Algebratodayisorganizedaxiomatically,andassuchitisabstract.Mathematiciansstudyalgebraicstructuresfromageneralpointofview,comparedifferentstructures,andfindrelationshipsbetweenthem.Thisabstractionandgeneralizationmightappeartobehopelesslyimpracticalbutitisnot!Thegeneralapproach in algebra has produced powerful new methods for algebraizing different parts ofmathematics and science, formulating problems which could never have been formulated before, andfindingentirelynewkindsofsolutions.

Suchexcursionsintopuremathematicalfancyhaveanoddwayofrunningaheadofphysicalscience,providing a theoretical framework to account for facts even before those facts are fully known. Thispattern is so characteristic that many mathematicians see themselves as pioneers in a world ofpossibilitiesratherthanfacts.Mathematiciansstudystructureindependentlyofcontent,andtheirscienceisavoyageofexplorationthroughallthekindsofstructureandorderwhichthehumanmindiscapableofdiscerning.

CHAPTER

TWOOPERATIONS

Addition, subtraction, multiplication, divisionthese and many others are familiar examples ofoperationsonappropriatesetsofnumbers.

Intuitively,anoperationonasetAisawayofcombininganytwoelementsofAtoproduceanotherelementinthesamesetA.

Everyoperationisdenotedbyasymbol,suchas+,,orInthisbookwewilllookatoperationsfromaloftyperspective;wewilldiscoverfactspertainingtooperationsgenerallyratherthantospecificoperationsonspecificsets.Accordingly,wewillsometimesmakeupoperationsymbolssuchas*andtorefertoarbitraryoperationsonarbitrarysets.

LetusnowdefineformallywhatwemeanbyanoperationonsetA.LetAbeanyset:

Anoperation*onAisarulewhichassignstoeachorderedpair(a,b)ofelementsofAexactlyoneelementa*binA.

Therearethreeaspectsofthisdefinitionwhichneedtobestressed:

1. a *b is defined for every ordered pair (a, b)of elements of A. There aremany ruleswhich lookdeceptivelylikeoperationsbutarenot,becausethisconditionfails.Oftena*b isdefinedforalltheobviouschoicesofaandb,butremainsundefinedinafewexceptionalcases.Forexample,divisiondoesnotqualifyasanoperationontheset oftherealnumbers,forthereareorderedpairssuchas(3,0)whosequotient3/0isundefined.Inordertobeanoperationon ,divisionwouldhavetoassociatearealnumberalbwitheveryorderedpair(a,b)ofelementsof .Noexceptionsallowed!

2. a*bmustbeuniquelydefined.Inotherwords,thevalueofa*bmustbegivenunambiguously.Forexample,onemightattempttodefineanoperationontheset oftherealnumbersbylettingabbethenumberwhosesquareisab.Obviouslythisisambiguousbecause28,letussay,maybeeither4or-4.Thus,doesnotqualifyasanoperationon !

3. IfaandbareinA,a*bmustbeinA.ThisconditionisoftenexpressedbysayingthatA isclosedundertheoperation*.Ifweproposetodefineanoperation*onasetA,wemusttakecarethat*,whenappliedtoelementsofA,doesnottakeusoutofA.Forexample,divisioncannotberegardedasanoperationonthesetoftheintegers,fortherearepairsofintegerssuchas(3,4)whosequotient3/4isnotaninteger.

Ontheotherhand,divisiondoesqualifyasanoperationonthesetofallthepositivereal

numbers,forthequotientofanytwopositiverealnumbersisauniquelydeterminedpositiverealnumber.

AnoperationisanyrulewhichassignstoeachorderedpairofelementsofAauniqueelementinA.Therefore it is obvious that there are, in general, many possible operations on a given setA. If, forexample,A is a set consisting of just twodistinct elements, saya andb, each operation onAmay bedescribedbyatablesuchasthisone:

IntheleftcolumnarelistedthefourpossibleorderedpairsofelementsofA,andtotherightofeachpair(x,y)isthevalueofx*y.Hereareafewofthepossibleoperations:

EachofthesetablesdescribesadifferentoperationonA.Eachtablehasfourrows,andeachrowmaybefilledwitheitheranaorab;hencethereare16possiblewaysoffillingthetable,correspondingto16possibleoperationsonthesetA.

WehavealreadyseenthatanyoperationonasetAcomeswithcertainoptions.Anoperation*maybecommutative,thatis,itmaysatisfy

a*b=b*a (1)

foranytwoelementsaandbinA.Itmaybeassociative,thatis,itmaysatisfytheequation

(a*b)*c=a*(b*c) (2)

foranythreeelementsa,b,andcinA.Tounderstandtheimportanceoftheassociativelaw,wemustrememberthatanoperationisawayof

combiningtwoelements;soifwewanttocombinethreeelements,wecandosoindifferentways.Ifwewanttocombinea,b,andcwithoutchangingtheirorder,wemayeithercombineawith theresultofcombiningbandc,whichproducesa*(b*c);orwemayfirstcombineawithb,andthencombinetheresult with c, producing (a * b) * c. The associative law asserts that these two possible ways ofcombiningthreeelements(withoutchangingtheirorder)producethesameresult.

Forexample,theadditionofrealnumbersisassociativebecausea+(b+c)=(a+b)+c.However,divisionofrealnumbersisnotassociative:forinstance,3/(4/5)is15/4,whereas(3/4)/5is3/20.

IfthereisanelementeinAwiththepropertythat

e*a=a and a*e=a foreveryelementainA (3)

then e is called an identity or neutral element with respect to the operation *. Roughly speaking,

Equation(3)tellsusthatwheneiscombinedwithanyelementa,itdoesnotchangea.Forexample,intheset oftherealnumbers,0isaneutralelementforaddition,and1isaneutralelementformultiplication.

IfaisanyelementofA,andxisanelementofAsuchthat

a*x=e and x*a=e (4)

thenxiscalledaninverseofa.Roughlyspeaking,Equation(4)tellsusthatwhenanelementiscombinedwithitsinverseitproducestheneutralelement.Forexample,intheset oftherealnumbers,a is theinverseofawithrespecttoaddition;ifa0,then1/aistheinverseofawithrespecttomultiplication.

The inverse of a is often denoted by the symbol al. (The symbol al is usually pronounced ainverse.)

EXERCISESThroughoutthisbook,theexercisesaregroupedintoexercisesets,eachsetbeingidentifiedbyaletterA,B,C, etc, andheadedby adescriptive title.Eachexercise set contains six to ten exercises, numberedconsecutively.Generally, the exercises in each set are independent of each other andmay be doneseparately.However,whentheexercisesinasetarerelated,withsomeexercisesbuildingonprecedingonessothattheymustbedoneinsequence,thisisindicatedwithasymboltinthemargintotheleftoftheheading.

Thesymbol#nexttoanexercisenumberindicatesthatapartialsolutiontothatexerciseisgivenintheAnswerssectionattheendofthebook.

A.ExamplesofOperationsWhichofthefollowingrulesareoperationsontheindicatedset?( designatesthesetoftheintegers,therationalnumbers,and therealnumbers.)Foreachrulewhichisnotanoperation,explainwhyitisnot.Example ,ontheset .

SolutionThisisnotanoperationon .Thereareintegersaandbsuchthat{a+b)/abisnotaninteger.(Forexample,

isnotaninteger.)Thus, isnotclosedunder*.1 ,ontheset .2 a*b=alnb,ontheset{x :x>0}.3 a*bisarootoftheequationx2a2b2=0,ontheset .4 Subtraction,ontheset .5 Subtraction,ontheset{n :0}.6 a*b=|ab,ontheset{n :0}.

B.PropertiesofOperationsEachofthefollowingisanoperation*onU.Indicatewhetherornot

(i) itiscommutative,(ii) itisassociative,(iii) hasanidentityelementwithrespectto*,(iv) everyx hasaninversewithrespectto*.

InstructionsFor(i),computex*yandy*x,andverifywhetherornottheyareequal.For(ii),computex*(y*z)and(x*y)*z,andverifywhetherornottheyareequal.For(iii),firstsolvetheequationx*e=x for e; if the equation cannot be solved, there is no identity element. If it can be solved, it is stillnecessarytocheckthate*x=x*e=xforanyx .Ifitchecks,theneisanidentityelement.For(iv),firstnote that if there isno identityelement, therecanbeno inverses. If there isan identityelemente, firstsolvetheequationx*x=eforxiftheequationcannotbesolved,xdoesnothaveaninverse.Ifitcanbesolved,checktomakesurethatx*x=x*x=x*x=e.Ifthischecks,xistheinverseofx.Examplex*y=x+y+1

(i) x*y=x+y+1;y*x=y+x+1=x+y+1.(Thus,*iscommutative.)

(ii) x*(y*z)=x*(y+z+l)=x+(y+z+l)+l=x+y+z+2.(x*y)*z=(x+y+1)*z=(x+y+1)+z+1=x+y+z+2.(*isassociative.)

(iii) Solvex*e=xfore:x*e=x+e+1=x;therefore,e=1.Check:x*(1)=x+(1)+1=x;(1)*x=(1)+x+1=x.Therefore,1istheidentityelement.(*hasanidentityelement.)

(iv) Solvex*x=1forx: x*x=x+x+1=1;thereforex=x2.Check:x*(x2)=x+(x2)+1=1;(x2)*x=(x2)+x+l=l.Therefore,x2istheinverseofx.(Everyelementhasaninverse.)

1 x*y=x+2y+4

(i) x*y=x+2y+4;y*x=(ii) x*(y*z)=x*( )=

(x*y)*z=( )*z=(iii) Solvex*e=xfore.Check.(iv) Solvex*x=eforx.Check.2 x*y=x+2yxy

3 x*y=|x+y|

4 x*y=|xy|

5 x*y=xy+1

6 x*y=max{x,y}=thelargerofthetwonumbersxandy

7 (onthesetofpositiverealnumbers)

C.OperationsonaTwo-ElementSetLetAbethetwo-elementsetA={a,b}.1 Writethetablesofall16operationsonA.(Usetheformatexplainedonpage20.)Labeltheseoperations0lto016.Then:

2 Identifywhichoftheoperations0lto016arecommutative.3 Identifywhichoperations,among0lto016,areassociative.4 Forwhichoftheoperations0lto016isthereanidentityelement?5 Forwhichoftheoperations0lto016doeseveryelementhaveaninverse?

D.Automata:TheAlgebraofInput/OutputSequencesDigital computers and related machines process information which is received in the form of inputsequences.AninputsequenceisafinitesequenceofsymbolsfromsomealphabetA.Forinstance,ifA={0,1}(thatis,ifthealphabetconsistsofonlythetwosymbols0and1),thenexamplesofinputsequencesare011010and10101111.IfA={a,b,c},thenexamplesofinputsequencesarebabbcacandcccabaa.Outputsequencesaredefinedinthesamewayasinputsequences.ThesetofallsequencesofsymbolsinthealphabetAisdenotedbyA*.

ThereisanoperationonA*calledconcatenation:IfaandbareinA*,saya=a1a2...anandb=blb2bm,then

ab=a1a2anbb2...bm

Inotherwords, thesequenceabconsistsof thetwosequencesaandbend toend.Forexample, in thealphabetA={0,1},ifa=1001andb=010,thenab=1001010.

Thesymboldenotestheemptysequence.1 Provethattheoperationdefinedaboveisassociative.2 Explainwhytheoperationisnotcommutative.3 Provethatthereisanidentityelementforthisoperation.

CHAPTER

THREETHEDEFINITIONOFGROUPS

Oneofthesimplestandmostbasicofallalgebraicstructuresisthegroup.Agroupisdefinedtobeasetwithanoperation(letuscallit*)whichisassociative,hasaneutralelement,andforwhicheachelementhasaninverse.Moreformally,

ByagroupwemeanasetGwithanoperation*whichsatisfiestheaxioms:(G1) *isassociative.(G2) ThereisanelementeinGsuchthata*e=aande*a=aforeveryelementainG.(G3) ForeveryelementainG,thereisanelementalinGsuchthata*a1=eanda1*a=e.

Thegroupwehave justdefinedmaybe representedby the symbol G, *. This notationmakes itexplicit that thegroupconsistsof thesetGand theoperation *. (Remember that, ingeneral, there areotherpossibleoperationsonG,soitmaynotalwaysbeclearwhichisthegroupsoperationunlessweindicateit.)Ifthereisnodangerofconfusion,weshalldenotethegroupsimplywiththeletterG.

Thegroupswhichcometomindmostreadilyarefoundinourfamiliarnumbersystems.Hereareafewexamples.

isthesymbolcustomarilyusedtodenotetheset

{,3,2,1,0,1,2,3,}

oftheintegers.Theset ,withtheoperationofaddition, isobviouslyagroup.It iscalled theadditivegroupoftheintegersandisrepresentedbythesymbol ,+.Mostly,wedenoteitsimplybythesymbol.

designatesthesetoftherationalnumbers(thatis,quotientsm/nofintegers,wheren0).Thisset,withtheoperationofaddition,iscalledtheadditivegroupoftherationalnumbers, ,+.Mostoftenwedenoteitsimplyby .

Thesymbol representsthesetoftherealnumbers. ,withtheoperationofaddition,iscalledtheadditivegroupoftherealnumbers,andisrepresentedby ,+,orsimply .

Thesetofall thenonzerorationalnumbers is representedby *.This set,with theoperationofmultiplication, is thegroup *, ,orsimply *.Similarly, thesetofall thenonzerorealnumbers isrepresentedby *.Theset *withtheoperationofmultiplication,isthegroup *,,orsimply *.

Finally, posdenotesthegroupofallthepositiverationalnumbers,withmultiplication. posdenotesthegroupofallthepositiverealnumbers,withmultiplication.

Groupsoccurabundantlyinnature.Thisstatementmeansthatagreatmanyofthealgebraicstructureswhichcanbediscerned innaturalphenomena turnout tobegroups.Typicalexamples,whichweshallexamine later, come up in connectionwith the structure of crystals, patterns of symmetry, and variouskinds of geometric transformations. Groups are also important because they happen to be one of thefundamentalbuildingblocksoutofwhichmorecomplexalgebraicstructuresaremade.

Especially important in scientific applications are the finite groups, that is, groups with a finitenumberofelements.Itisnotsurprisingthatsuchgroupsoccurofteninapplications,forinmostsituationsoftherealworldwedealwithonlyafinitenumberofobjects.

Theeasiestfinitegroupstostudyarethosecalledthegroupsofintegersmodulon(wherenisanypositive integer greater than 1).These groupswill be described in a casualway here, and a rigoroustreatmentdeferreduntillater.

Letusbeginwithaspecificexample,say,thegroupofintegersmodulo6.Thisgroupconsistsofasetofsixelements,

{0,1,2,3,4,5}

andanoperationcalledadditionmodulo6,whichmaybedescribedasfollows:Imaginethenumbers0through5asbeingevenlydistributedonthecircumferenceofacircle.Toaddtwonumbershandk,startwithhandmoveclockwisekadditionalunitsaroundthecircle:h+kiswhereyouendup.Forexample,3+3=0, 3+5=2, and soon.The set {0, 1, 2, 3, 4, 5}with thisoperation is called thegroup ofintegersmodulo6,andisrepresentedbythesymbol 6.

Ingeneral,thegroupofintegersmodulonconsistsoftheset

{0,1,2,,n1}

with the operation of additionmodulo n, which can be described exactly as previously. Imagine thenumbers0 throughn1 tobepointson theunitcircle,eachoneseparated fromthenextbyanarcoflength2/n.

Toaddhandk, startwithh andgoclockwise throughanarcofk times2/n.The sumh +kwill, of

course,beoneofthenumbers0throughn1.Fromgeometricalconsiderationsitisclearthatthiskindofaddition (by successive rotations on the unit circle) isassociative. Zero is the neutral element of thisgroup,andnhisobviouslytheinverseofh[forh+(nh)=n,whichcoincideswith0].Thisgroup,thegroupofintegersmodulon,isrepresentedbythesymbol n.

Oftenwhenworkingwithfinitegroups,itisusefultodrawupanoperationtable.Forexample,theoperationtableof 6is

Thebasicformatofthistableisasfollows:

withonerowforeachelementofthegroupandonecolumnforeachelementofthegroup.Then3+4,forexample,islocatedintherowof3andthecolumnof4.Ingeneral,anyfinitegroupG,*hasatable

Theentryintherowofxandthecolumnofyisx*y.Let us remember that the commutative law is not one of the axioms of group theory; hence the

identitya*b=b*aisnottrueineverygroup.IfthecommutativelawholdsinagroupG,suchagroupiscalledacommutativegroupor,morecommonly,anabeliangroup.Abeliangroupsarenamedafter themathematician Niels Abel, who was mentioned in Chapter 1 and who was a pioneer in the study ofgroups.Alltheexamplesofgroupsmentioneduptonowareabeliangroups,buthereisanexamplewhichisnot.

LetGbethegroupwhichconsistsofthesixmatrices

withtheoperationofmatrixmultiplicationwhichwasexplainedonpage8.Thisgrouphasthefollowingoperationtable,whichshouldbechecked:

Inlinearalgebraitisshownthatthemultiplicationofmatricesisassociative.(Thedetailsaresimple.)ItisclearthatIistheidentityelementofthisgroup,andbylookingatthetableonecanseethateachofthesixmatricesin{I,A,B,C,D,K}hasaninversein{I,A,B,C,D,K}.(Forexample,BistheinverseofD,AistheinverseofA,andsoon.)Thus,Gisagroup!NowweobservethatAB=CandBA=K,soGisnotcommutative.

EXERCISES

A.ExamplesofAbelianGroupsProvethateachofthefollowingsets,withtheindicatedoperation,isanabeliangroup.

InstructionsProceedasinChapter2,ExerciseB.1x*y=x+y+k (kafixedconstant),ontheset oftherealnumbers.2 ,ontheset{x :x0}.3x*y=x+y+xy,ontheset{x :x1}.4 ,theset{x :10}. His isnot asubgroupofG.3G= ,+,H={x :tanx }. His isnot asubgroupofG.

HINT:Usethefollowingformulafromtrigonometry:

4G= *,,H={2n3m:m,n }. His isnot asubgroupofG.5G= ,+,H={(x,y):y=2x}. His isnot asubgroupofG.6G= ,+,H={(x,y):x2+y2>0}. His isnot asubgroupofG.7LetCandDbesets,withCD.ProvethatPCisasubgroupofPD.(SeeChapter3,ExerciseC.)

B.SubgroupsofFunctionsIneachofthefollowing,showthatHisasubgroupofG.

ExampleG= ( ),+,H={f ( ):f(0)=0}(i)Supposef,gH;thenf(0)=0andg(0)=0,so[f+g](0)=f(0)+g(0)=0+0=0.Thus,f+g

H.(ii)IffH,thenf(0)=0.Thus,[f](0)=f(0)=0=0,sofH.

1G= ( ),+,H={f ( ):f(x)=0foreveryx[0,1]}2G= ( ),+,H={f ( ):f(x)=f(x)}3G= ( ),+,H={f ( ):fisperiodicofperiod}REMARK:Afunctionfissaidtobeperiodicofperiodaifthereisanumbera,calledtheperiodoff,suchthatf(x)=f(x+na)foreveryx andn .4G= ( ),+,H={f ( ):5G= ( ),+,H={f ( ):df/dxisconstant}6G= ( ),+,H={f ( ):f(x) foreveryx }

C.SubgroupsofAbelianGroupsInthefollowingexercises,letGbeanabeliangroup.1IfH={xG:x=x1},thatis,HconsistsofalltheelementsofGwhicharetheirowninverses,provethatHisasubgroupofG.2Letnbeafixedinteger,andletH={xG:xn=e}.ProvethatHisasubgroupofG.3LetH={xG:x=y2forsomeyG};thatis,letHbethesetofalltheelementsofGwhichhaveasquareroot.ProvethatHisasubgroupofG.4LetHbeasubgroupofG,andletK={xG:x2H}.ProvethatKisasubgroupofG.#5.LetHbeasubgroupofG,andletKconsistofalltheelementsxinGsuchthatsomepowerofxisinH.Thatis,K={xG:forsomeintegern>0,xnH).ProvethatKisasubgroupofG.6SupposeHandKaresubgroupsofG,anddefineHKasfollows:

HK={xy:xHandyK}

ProvethatHKisasubgroupofG.7Explainwhyparts46arenottrueifGisnotabelian.

D.SubgroupsofanArbitraryGroupLetGbeagroup.1IfHandKaresubgroupsofagroupG,provethatHKisasubgroupofG.(RememberthatxHKiffxHandxK.)2LetHandKbesubgroupsofG.ProvethatifHK,thenHisasubgroupofK.3BythecenterofagroupGwemeanthesetofalltheelementsofGwhichcommutewitheveryelementofG,thatis,

C={aG:ax=xaforeveryxG)

ProvethatCisasubgroupofG.4LetC={aG:(ax)2=(xa)2foreveryxG).ProvethatCisasubgroupofG.#5LetGbea finitegroup,and letSbeanonemptysubsetofG.SupposeS isclosedwithrespect tomultiplication. Prove thatS is a subgroup ofG. (HINT: It remains to prove thatS contains e and isclosedwithrespecttoinverses.LetS={a1,,an}.IfaiS,considerthedistinctelementsaia1,aia2,,aian.)

6LetGbeagroupandf:GGafunction.AperiodoffisanyelementainGsuchthatf(x)=f(ax)foreveryxG.Prove:ThesetofalltheperiodsoffisasubgroupofG.#7LetHbeasubgroupofG,andletK={xG:xax1HiffaH}.Prove:

(a)KisasubgroupofG.(b)HisasubgroupofK.

8LetGandHbegroups,andGHtheirdirectproduct.(a)Provethat{(x,e):xG}isasubgroupofGH.(b)Provethat{(x,x):xG}isasubgroupofGG.

E.GeneratorsofGroups1Listallthecyclicsubgroupsof 10,+.2Showthat 10isgeneratedby2and5.3Describethesubgroupof 12generatedby6and9.4Describethesubgroupof generatedby10and15.5Showthat isgeneratedby5and7.6Showthat 2 3isacyclicgroup.Showthat 3 4isacyclicgroup.#7Showthat 2 4isnotacyclicgroup,butisgeneratedby(1,1)and(1,2).8SupposeagroupGisgeneratedbytwoelementsaandb.Ifab=ba,provethatGisabelian.

F.GroupsDeterminedbyGeneratorsandDefiningEquations#1LetGbethegroup{e,a,b,b2,ab,ab2}whosegeneratorssatisfya2=e,b3=e,ba=ab2.WritethetableofG.

2LetGbethegroup{e,a,b,b2,b3,ab,ab2,ab3}whosegeneratorssatisfya2=e,b4=e,ba=ab3.WritethetableofG.(GiscalledthedihedralgroupD4.)

3LetGbethegroup{e,a,b,b2,b3,ab,ab2,ab3}whose,generatorssatisfya4=e,a2=b2,ba=ab3.WritethetableofG.(Giscalledthequaterniongroup.)4LetGbethecommutativegroup{e,a,b,c,ab,be,ac,abc}whosegeneratorssatisfya2=b2=c2=e.WritethetableofG.

G.CayleyDiagramsEvery finite groupmay be represented by a diagram known as a Cayley diagram. A Cayley diagramconsistsofpointsjoinedbyarrows.

Thereisonepointforeveryelementofthegroup.Thearrowsrepresenttheresultofmultiplyingbyagenerator.

For example, ifG has only one generator a (that is,G is the cyclic group a), then the arrow representstheoperationmultiplybya:

eaa2a3

Ifthegrouphastwogenerators,sayaandb,weneedtwokindsofarrows,say and,where meansmultiplybya,andmeansmultiplybyb.

Forexample,thegroupG={e,a,b,b2,ab,ab2}wherea2=e,b3=e,andba=ab2(seepage47)hasthefollowingCayleydiagram:

Movingintheforwarddirectionofthearrowmeansmultiplyingbyb,

whereasmovinginthebackwarddirectionofthearrowmeansmultiplyingbyb1:

(Notethatmultiplyingxbybisunderstoodtomeanmultiplyingontherightbyb:itmeansxb,notbx.)Itisalsoaconventionthatifa2=e(hencea=a1),thennoarrowheadisused:

forifa=a1,thenmultiplyingbyaisthesameasmultiplyingbya1.TheCayleydiagramofagroupcontainsthesameinformationasthegroupstable.Forinstance,to

findtheproduct(ab)(ab2)inthefigureonpage51,westartatabandfollowthepathcorrespondingtoab2(multiplyingbya,thenbyb,thenagainbyb),whichis

Thispathleadstob;hence(ab)(ab2)=b.Asanotherexample,theinverseofab2isthepathwhichleadsfromab2backtoe.Wenoteinstantly

thatthisisba.Apoint-and-arrowdiagramistheCayleydiagramofagroupiffithasthefollowingtwoproperties:

(a)Foreachpointxandgeneratora,thereisexactlyonea-arrowstartingatx,andexactlyonea-arrowending atx; furthermore, atmost one arrow goes from x to another point y. (b) If two different pathsstartingatx lead to thesamedestination, then these twopaths,startingatanypointy, lead to thesamedestination.

Cayleydiagramsareausefulwayoffindingnewgroups.Write the tableof thegroupshaving the followingCayleydiagrams: (REMARK:Youmay take any

point to represent e, because there is perfect symmetry in aCayley diagram.Choose e, then label thediagramandproceed.)

H.CodingTheory:GeneratorMatrixandParity-CheckMatrixofaCodeForthereaderwhodoesnotknowthesubject,linearalgebrawillbedevelopedinChapter28.However,somerudimentsofvectorandmatrixmultiplicationwillbeneededinthisexercise;theyaregivenhere:

Avectorwithncomponents isasequenceofnnumbers: (a1,a2,,an).Thedotproductof twovectorswithncomponents,saythevectorsa=(a1,a2,,an)andb=(b1,b2,,bn),isdefinedby

ab=(a1,a2,,an)(b1,b2,,bn)=a1b1+a2b2++anbn

thatis,youmultiplycorrespondingcomponentsandadd.Forexample,

(1,4,2,3)(6,2,4,2)=1(6)+4(2)+(2)4+3(2)=0

Whenab=0,asinthelastexample,wesaythataandbareorthogonal.Amatrix isarectangulararrayofnumbers.Anmbynmatrix(mnmatrix)hasm rowsandn

columns.Forexample,

isa34matrix: Ithas threerowsandfourcolumns.Notice thateachrowofB is avectorwith fourcomponents,andeachcolumnofBisavectorwiththreecomponents.

IfAisanymnmatrix,leta1,a2,,anbethecolumnsofA.(EachcolumnofAisavectorwithmcomponents.)Ifxisanyvectorwithmcomponents,xAdenotesthevector

xA=(xa1,xa2,,xan)

That is, the componentsofxA are obtainedbydotmultiplyingx by the successive columnsofA. Forexample,ifBisthematrixofthepreviousparagraphandx=(3,1,2),thenthecomponentsofxBare

thatis,xB=(7,3,15,8).IfAisanmnmatrix,leta(1),a(2),,a(m)betherowsofA.Ifyisanyvectorwithncomponents,

Aydenotesthevector

Ay=(ya(1),ya(2),,ya(m))

Thatis,thecomponentsofAyareobtainedbydotmultiplyingywiththesuccessiverowsofA.(Clearly,AyisnotthesameasyA.)Fromlinearalgebra,A(x+y)=Ax+Ayand(A+B)x=Ax+Bx.

WeshallnowcontinuethediscussionofcodesbeguninExercisesFandGofChapter3.Recallthatisthesetofallvectorsoflengthnwhoseentriesare0sand1s.InExerciseF,page32,itwasshown

that isagroup.AcodeisdefinedtobeanysubsetCof .Acodeiscalledagroupcode ifC isasubgroupof .ThecodesdescribedinChapter3,aswellasallthosetobementionedinthisexercise,aregroupcodes.

AnmnmatrixGisageneratormatrixforthecodeCifCisthegroupgeneratedbytherowsofG.Forexample,ifC1isthecodegivenonpage34,itsgeneratormatrixis

YoumaycheckthatalleightcodewordsofC1aresumsoftherowsofG1.Recallthateverycodewordconsistsofinformationdigitsandparity-checkdigits.InthecodeC1the

firstthreedigitsofeverycodewordareinformationdigits,andmakeupthemessage;thelasttwodigitsareparity-checkdigits.Encodingamessageistheprocessofaddingtheparity-checkdigitstotheendofthemessage.Ifx isamessage, thenE(x)denotes theencodedword.Forexample, recall that inC1 theparity-check equations area4 =a1 +a3 anda5 = a1 +a2 +a3. Thus, a three-digitmessagea1a2a3 isencodedasfollows:

E(a1,a2,a3)=(a1,a2,a3,a1+a3,a1+a2+a3)

Thetwodigitsaddedattheendofawordarethosedictatedbytheparitycheckequations.Youmayverifythat

Thisistrueinallcases:IfGisthegeneratormatrixofacodeandxisamessage,thenE(x)isequaltotheproductxG.Thus,encodingusingthegeneratormatrixisveryeasy:yousimplymultiplythemessagexbythegeneratormatrixG.

Now,theparity-checkequationsofC1(namely,a4=a1+a3anda5=a1+a2+a3)canbewrittenintheform

a1+a3+a4=0 and a1+a2+a3+a5=0

whichisequivalentto

(a1,a2,a3,a4,a5)(1,0,1,1,0)=0

and

(a1,a2,a3,a4,a5)(1,1,1,0,1)=0

The last two equations show that aworda1a2a3a4a5 is a codeword (that is, satisfies the parity-checkequations)ifandonlyif(a1,a2,a3,a4,a5)isorthogonaltobothrowsofthematrix:

Hiscalledtheparity-checkmatrixofthecodeC1.Thisconclusionmaybestatedasatheorem:Theorem1LetHbetheparity-checkmatrixofacodeCin .Awordx in isacodeword if

andonlyifHx=0.(RememberthatHxisobtainedbydotmultiplyingxbytherowsofH.)1FindthegeneratormatrixG2andtheparity-checkmatrixH2ofthecodeC2describedinExerciseG2ofChapter3.2LetC3be the followingcode in : the first fourpositionsare informationpositions,and theparity-checkequationsarea5=a2+a3+a4,a6=a1+a3+a4,anda7=a1+a2+a4.(C3iscalledtheHammingcode.)FindthegeneratormatrixG3andparity-checkmatrixH3ofC3.

Theweight of a word x is the number of Is in the word and is denoted byw(x). For example,w(11011)=4.TheminimumweightofacodeCistheweightofthenonzerocodewordofsmallestweightinthecode.(Seethedefinitionsofdistanceandminimumdistanceonpage34.)Provethefollowing:#3d(x,y)=w(x+y).

4w(x)=d(x,0),where0isthewordwhosedigitsareall0s.5TheminimumdistanceofagroupcodeCisequaltotheminimumweightofC.6(a)Ifxandyhaveevenweight,sodoesx+y.

(b)Ifxandyhaveoddweight,x+yhasevenweight.(c)Ifxhasoddandyhasevenweight,thenx+yhasoddweight.

7Inanygroupcode,eitherallthewordshaveevenweight,orhalfthewordshaveevenweightandhalfthewordshaveoddweight.(Usepart6inyourproof.)8H(x+y)=0ifandonlyifHx=Hy,whereHdenotestheparity-checkmatrixofacodein andxandyareanytwowordsin ).

CHAPTER

SIX

FUNCTIONS

The concept of a function is one of the most basic mathematical ideas and enters into almost everymathematicaldiscussion.Afunctionisgenerallydefinedasfollows:IfAandBaresets,thenafunctionfromA toB is a rulewhich to every elementx inA assigns aunique elementy inB. To indicate thisconnectionbetweenxandyweusuallywritey=f(x),andwecallytheimageofxunderthefunctionf.

Thereisnothinginherentlymathematicalaboutthisnotionoffunction.Forexample,imagineAtobeasetofmarriedmenandBtobethesetoftheirwives.Letfbetherulewhichtoeachmanassignshiswife.ThenfisaperfectlygoodfunctionfromAtoB;underthisfunction,eachwifeistheimageofherhusband.(Nopunisintended.)

Takecare,however,tonotethatiftherewereabachelorinAthenfwouldnotqualifyasafunctionfromAtoB;forafunctionfromAtoBmustassignavalueinBtoeveryelementofA,withoutexception.Now,supposethemembersofAandBareAshanti,amongwhompolygamyiscommon;inthelandoftheAshanti, fdoesnotnecessarilyqualifyasa function, for itmayassign toagivenmemberofA severalwives.If/isafunctionfromAtoB,itmustassignexactlyoneimagetoeachelementofA.

IffisafunctionfromAtoBitiscustomarytodescribeitbywriting

f:AB

ThesetAiscalledthedomainoff.TherangeoffisthesubsetofBwhichconsistsofalltheimagesofelementsofA.Inthecaseofthefunctionillustratedhere,{a,b,c}isthedomainoff,and{x,y}isthe

rangeof f(z is not in the range of f). Incidentally, this function fmay be represented in the simplifiednotation

ThisnotationisusefulwheneverAisafiniteset:theelementsofAarelistedinthetoprow,andbeneatheachelementofAisitsimage.

Itmayperfectlywellhappen,iffisafunctionfromAtoB,thattwoormoreelementsofAhavethesameimage.Forexample, ifwe lookat thefunction immediatelyabove,weobserve thataandbbothhavethesameimagex.Iffisafunctionforwhichthiskindofsituationdoesnotoccur,thenfiscalledaninjectivefunction.Thus,

Definition1Afunctionf:ABiscalledinjectiveifeachelementofBistheimageofnomorethanoneelementofA.

Theintendedmeaning,ofcourse,isthateachelementyinBistheimageofnotwodistinctelementsofA.Soif

that is, xl and x2 have the same image y, wemust require that xl be equal to x2. Thus, a convenientdefinitionofinjectiveisthis:afunctionf:ABisinjectiveifandonlyif

f(x1)=f(x2) implies x1=x2

IffisafunctionfromAtoB,theremaybeelementsinBwhicharenotimagesofelementsofA. Ifthisdoesnothappen, that is, ifeveryelementofB is the imageofsomeelementofA,wesay that f issurjective.

Definition2Afunctionf:ABiscalledsurjectiveifeachelementofBistheimageofatleastoneelementofA.

ThisisthesameassayingthatBistherangeoff.Now,supposethatfisbothinjectiveandsurjective.ByDefinitions1and2,eachelementofBisthe

imageofatleastoneelementofA,andnomorethanoneelementofA.SoeachelementofBistheimageofexactlyoneelementofA.Inthiscase,fiscalledabijectivefunction,oraone-to-onecorrespondence.

Definition3Afunctionf:ABiscalledbijectiveifitisbothinjectiveandsurjective.

It isobviousthatunderabijectivefunction,eachelementofAhasexactlyonepartnerinBandeachelementofBhasexactlyonepartnerinA.

Themost naturalwayofcombining two functions is to form their composite.The idea is this:supposefisafunctionfromAtoB,andgisafunctionfromBtoC.WeapplyftoanelementxinAandget anelementy inB; thenwe applyg to y andget an element z inC.Thus, z is obtained from x byapplyingfandginsuccession.Thefunctionwhich

consistsofapplyingfandginsuccessionisafunctionfromAtoC,andiscalledthecompositeoffandg.Moreprecisely,

Letf:ABandg:BCbefunctions.ThecompositefunctiondenotedbygfisafunctionfromAtoCdefinedasfollows:

[gf](x)=g(f(x)) foreveryxA

Forexample,consideronceagainasetAofmarriedmenandthesetBoftheirwives.LetCbethesetofallthemothersofmembersofB.Letf:ABbetherulewhichtoeachmemberofAassignshiswife,andg:BCtherulewhichtoeachwomaninBassignshermother.Thengf is themother-in-lawfunction,whichassignstoeachmemberofAhiswifesmother:

Foranother,moreconventional,example,letfandgbethefollowingfunctionsfrom to :f(x)=2x;g(x)=x+1.(Inotherwords,fistherulemultiplyby2andgistheruleadd1.)Theircompositesarethefunctionsgfandfggivenby

[fg](x)=f(g(x))=2(x+1)

and

[gf](x)=g(f(x))=2x+1

fgandgfaredifferent:fgistheruleadd1,thenmultiplyby2,whereasgfistherulemultiplyby2andthenadd1.

Itisanimportantfactthatthecompositeoftwoinjectivefunctionsisinjective,thecompositeoftwosurjective functions is surjective, and the composite of two bijective functions is bijective. In otherwords,iff:ABandg:BCarefunctions,thenthefollowingaretrue:

Iffandgareinjective,thengfisinjective.Iffandgaresurjective,thengfissurjective.Iffandgarebijective,thengfisbijective.

Letustackleeachoftheseclaimsinturn.Wewillsupposethatfandgareinjective,andprovethatgfisinjective.(Thatis,wewillprovethatif[gf](x)=[gf](y),thenx=y.)

Suppose[gf](x)=[gf](y),thatis,

g(f(x))=g(f(y))

Becausegisinjective,weget

f(x)=f(y)

andbecausefisinjective,

x=y

Next,letussupposethatfandgaresurjective,andprovethatgfissurjective.Whatweneedtoshowhereis thateveryelementofC isg fofsomeelementofA.Well, ifzC, then(becauseg issurjective)x=g(y)forsomeyB;butfissurjective,soy=f(x)forsomexA.Thus,

z=g(y)=g(f(x))=[gf](x)

Finally,iffandgarebijective,theyarebothinjectiveandsurjective.Bywhatwehavealreadyproved,gfisinjectiveandsurjective,hencebijective.

AfunctionffromAtoBmayhaveaninverse,butitdoesnothaveto.Theinverseoff,ifitexists,isa

functionf1(finverse)fromBtoAsuchthat

x=f1(y) ifandonlyif y=f(x)

Roughlyspeaking,iffcarriesxtoythenflcarriesytox.Forinstance,returning(forthelasttime)totheexampleofasetAofhusbandsandthesetBoftheirwives,iff:ABistherulewhichtoeachhusbandassignshiswife,thenfl:BAistherulewhichtoeachwifeassignsherhusband:

Ifwethinkoffunctionsasrules,thenf1istherulewhichundoeswhateverfdoes.Forinstance,iffisthe real-valued function f(x)=2x, then f1 is the function f1(x)=x/2 [or, if preferred, f1(y) = y/2].Indeed,theruledivideby2undoeswhattherulemultiplyby2does.

Whichfunctionshaveinverses,andwhichothersdonot?Iff,afunctionfromAtoB,isnotinjective,itcannothaveaninverse;fornotinjectivemeansthereareatleasttwodistinctelementsx1andx2withthesameimagey;

Clearly,x1=f1(y)andx2= f1(y)so fl(y) isambiguous (ithas twodifferentvalues),and this isnotallowedforafunction.

Iff,afunctionfromAtoBisnotsurjective,thereisanelementyinBwhichisnotanimageofanyelementofA;thusfl(y)doesnotexist.Sof1cannotbeafunctionfromB(thatis,withdomainB)toA.

Itisthereforeobviousthatifflexists,fmustbeinjectiveandsurjective,thatis,bijective.Ontheotherhand,iffisabijectivefunctionfromAtoB,itsinverseclearlyexistsandisdeterminedbytherulethatify=f(x)thenf1(y)=x.

Furthermore,itiseasytoseethattheinverseoffisalsoabijectivefunction.Tosumup:Afunctionf:ABhasaninverseifandonlyifitisbijective.Inthatcase,theinverseflisabijectivefunctionfromBtoA.

EXERCISES

A.ExamplesofInjectiveandSurjectiveFunctionsEachofthefollowingisafunctionf: .Determine

(a)whetherornotfisinjective,and(b)whetherornotfissurjective.

Proveyouranswerineithercase.

Example1f(x)=2x

fisinjective.PROOFSupposef(a)=f(b),thatis,

2a=2b

Then

a=b

Thereforefisinjective.fissurjective.PROOFTakeanyelementy .Theny=2(y/2)=f(y/2).Thus,everyy isequaltof(x)forx=y/2.Thereforefissurjective.

Example2f(x)=x2

fisnotinjective.PROOFByexhibitingacounterexample:f(2)=4f(2),although22.fisnotsurjective.PROOFByexhibitingacounterexample:1isnotequaltof(x)foranyx .

1 f(x)=3x+42 f(x)=x3+1

3 f(x)=|x|#4 f(x)=x33x

5

#67 Determinetherangeofeachofthefunctionsinparts1to6.

B.Functionson andDeterminewhethereachofthefunctionslistedinparts14isorisnot(a)injectiveand(b)surjective.ProceedasinExerciseA.1 f: (0,),definedbyf(x)=ex.2 f:(0,1),definedbyf(x)=tanx.3 f: ,definedbyf(x)=theleastintegergreaterthanorequaltox.4 f: ,definedby

5 Findabijectivefunctionffromtheset oftheintegerstothesetEoftheevenintegers.

C.FunctionsonArbitrarySetsandGroupsDeterminewhethereachofthefollowingfunctionsisorisnot(a)injectiveand(b)surjective.ProceedasinExerciseA.

Inparts1to3,AandBaresets,andABdenotesthesetofalltheorderedpairs(x,y)asxrangesoverAandyoverB.1 f:ABA,definedbyf(x,y)=x.2 f:ABBA,definedbyf(x,y)=(y,x).3 f:AB,definedbyf(x)=(x,b),wherebisafixedelementofB.4 Gisagroup,aG,andf:GGisdefinedbyf(x)=ax.5 Gisagroupandf:GGisdefinedbyf(x)=x1.6 Gisagroupandf:GGisdefinedbyf(x)=x2.

D.CompositeFunctionsInparts13findthecompositefunction,asindicated.1 f: isdefinedbyf(x)=sinx.g: isdefinedbyg(x)=ex.

Findfgandgf.2 AandBaresets;f:ABBAisgivenbyf(x,y)=(y,x).g:BABisgivenbyg(y,x)=y.

Findgf.3 f:(0,1) isdefinedbyf(x)=1/x.g: isdefinedbyg(x)=Inx.

Findgf.Explainwhyfgisundefined.

4 Inschool,JackandSamexchangednotesinacodefwhichconsistsofspellingeverywordbackwardsandinterchangingeveryletterswitht.Alternatively, theyuseacodegwhichinterchangesthelettersawitho,iwithu,ewithy,andswitht.Describethecodesfgandgf.Aretheythesame?5 A={a,b,c,d};fandgarefunctionsfromAtoA;inthetabularformdescribedonpage57,theyaregivenby

Givefgandgfinthesametabularform.6 Gisagroup,andaandbareelementsofG.f:GGisdefinedbyf(x)=ax.g:GGisdefinedbyg(x)=bx.

Findfgandgf.7 Indicatethedomainandrangeofeachofthecompositefunctionsyoufoundinparts1to6.

E.InversesofFunctionsEachofthefollowingfunctionsfisbijective.Describeitsinverse.1f:(0,)(0,),definedbyf(x)=1/x.2f: (0,),definedbyf(x)=ex.3f: ,definedbyf(x)=x3+1.4f: ,definedby

5A={a,b,c,d},B={1,2,3,4}andf:ABisgivenby

6Gisagroup,aG,andf:GGisdefinedbyf(x)=ax.

F.FunctionsonFiniteSets1ThemembersoftheU.N.PeaceCommitteemustchoose,fromamongthemselves,apresidingofficeroftheir committee. For each member x, let f(x) designate that members choice for officer. If no twomembersvotealike,whatistherangeoff?2LetAbeafiniteset.Explainwhyanyinjectivefunctionf:AA isnecessarilysurjective. (Lookatpart1.)3IfAisafiniteset,explainwhyanysurjectivefunctionf:AAisnecessarilyinjective.4Arethestatementsinparts2and3truewhenAisaninfiniteset?Ifnot,giveacounterexample.#5IfAhasnelements,howmanyfunctionsaretherefromAtoA?HowmanybijectivefunctionsaretherefromAtoA?

G.SomeGeneralPropertiesofFunctionsInparts1to3,letA,B,andCbesets,andletf:ABandg:BCbefunctions.1Provethatifgfisinjective,thenfisinjective.

2Provethatifgfissurjective,thengissurjective.3Parts1and2, together, tellus that ifg f isbijective, then f is injectiveandg is surjective. Is theconverseofthisstatementtrue:Ifxy0isinjectiveandgsurjective,isgfbijective?(Ifyes,proveit;ifno,giveacounterexample.)4Letf:ABandg:BAbefunctions.Supposethaty=f(x)iffx=g(y).Provethatfisbijective,andg=f1

H.TheoryofAutomataDigitalcomputersandotherelectronicsystemsaremadeupofcertainbasiccontrolcircuits.Underlyingsuchcircuits isafundamentalmathematicalnotion, thenotionof finiteautomata,alsoknownas finite-statemachines.

AfiniteautomatonreceivesinformationwhichconsistsofsequencesofsymbolsfromsomealphabetA.Atypicalinputsequenceisawordx=x1x2xn,wherexlx2,...aresymbolsinthealphabetA.Themachine has a set of internal components whose combined state is called the internal state of themachine. At each time interval the machine reads one symbol of the incoming input sequence andrespondsbygoingintoanewinternalstate:theresponsedependsbothonthesymbolbeingreadandonthemachinespresentinternalstate.LetSdenotethesetofinternalstates;wemaydescribeaparticularmachinebyspecifyingafunction:SAS. Ifsi isan internalstateandaj is thesymbolcurrentlybeingread,then(siaj)=skisthemachinesnextstate.(Thatis,themachine,whileinstatesirespondstothe symbol aj by going into the new state sk.) The function is called thenext-state function of themachine.

ExampleLetM1bethemachinewhosealphabetisA={0,1},whosesetofinternalstatesisS={s0,s1},andwhosenext-statefunctionisgivenbythetable

(Thetableasserts:Wheninstates0andreading0,remainins0.Whenins0andreading1,gotostates1.Whenins1andreading0,remainins1Whenins1andreading1,gotostates0.)

Apossible useofM1 is as aparity-checkmachine,whichmay be used in decoding informationarriving on a communication channel. Assume the incoming information consists of sequences of fivesymbols, 0s and Is, suchas10111.Themachine startsoff in state s0. It reads the first digit, 1, and asdictatedbythetableabove,goesintostates1.Thenitreadstheseconddigit,0,andremainsins1.Whenitreadsthethirddigit,1,itgoesintostates0,andwhenitreadsthefourthdigit,1,itgoesintos1.Finally,itreachesthelastdigit,whichistheparity-checkdigit:ifthesumofthefirstfourdigitsiseven,theparity-checkdigitis0;otherwiseitis1.Whentheparity-checkdigit,1,isread,themachinegoesintostates0.Endinginstates0indicatesthattheparity-checkdigitiscorrect.Ifthemachineendsins1,thisindicatesthattheparity-checkdigitisincorrectandtherehasbeenanerroroftransmission.

A machine can also be described with the aid of a state diagram, which consists of circlesinterconnectedbyarrows:thenotation

meansthatifthemachineisinstatesiwhenxisread,itgoesintostatesj.ThediagramofthemachineM1ofthepreviousexampleis

Inparts 14 describe themachineswhich are able to carry out the indicated functions. For eachmachine,givethealphabetA, thesetofstatesS,and the tableof thenext-state function.Thendrawthestatediagramofthemachine.#1Theinputalphabetconsistsoffourletters,a,b,c,andd.Foreach incomingsequence, themachinedetermineswhetherthatsequencecontainsexactlythreeas.

2Thesameconditionspertainasinpart1,butthemachinedetermineswhetherthesequencecontainsatleastthreeas.3Theinputalphabetconsistsofthedigits0,1,2,3,4.Themachineaddsthedigitsofaninputsequence;theadditionismodulo5(seepage27).Thesumistobegivenbythemachinesstateafterthelastdigitisread.4Themachinetellswhetherornotanincomingsequenceof0sandIsendswith111.5IfMisamachinewhosenext-statefunctionis,defineasfollows:Ifxisaninputsequenceandthemachine(instatesi.)beginsreadingx,then(si,x)isthestateofthemachineafterthelastsymbolofxisread.Forinstance,ifM1isthemachineoftheexamplegivenabove,then(s0,11010)=s1.(Themachineisins0beforethefirstsymbolisread;each1altersthestate,butthe0sdonot.Thus,afterthelast0isread,themachineisinstates1.)(a)ForthemachineM1,give(s0,x)forallthree-digitsequencesx.(b)Forthemachineofpart1,give(si,x)foreachstatesiandeverytwo-lettersequencex.6WitheachinputsequencexweassociateafunctionTx:SScalledastatetransitionfunction,definedasfollows:

Tx(si)=(si,x)

ForthemachineM1oftheexample,ifx=11010,Txisgivenby

Tx(s0)=s1 and Tx(s1)=s0

(a)Describe the transition functionTx for themachineM1 and the following sequences:x = 01001,x=10011,x=01010.(b)ExplainwhyM1hasonlytwodistinct transitionfunctions.[Note:Twofunctions fandgareequal iff(x)=g(x)foreveryx;otherwisetheyaredistinct.](c) For themachine of part 1, describe the transition functionT1 for the followingx:x =abbca,x =

babac,x=ccbaa.(d)Howmanydistincttransitionfunctionsarethereforthemachineofpart3?

I.Automata,Semigroups,andGroupsByasemigroupwemeanasetAwithanassociativeoperation. (Theredoesnotneedtobean identityelement,nordoelementsnecessarilyhaveinverses.)Everygroupisasemigroup,thoughtheconverseisclearlyfalse.WitheverysemigroupAweassociateanautomatonM=M(A)calledtheautomatonofthesemigroupA.ThealphabetofMisA,thesetofstatesalsoisA,andthenext-statefunctionis(s,a)=sa[or(s,a)=s+aiftheoperationofthesemigroupisdenotedadditively].1DescribeM( 4).Thatis,givethetableofitsnext-statefunction,aswellasitsstatediagram.2DescribeM(S3).

IfM is amachineandS is the setof statesofM, thestate transition functions ofM (defined inExerciseH6ofthischapter)arefunctionsfromStoS.InthenextexerciseyouwillbeaskedtoshowthatTy Tx = Txy; that is, the composite of two transition functions is a transition function. Since thecompositionoffunctionsisassociative[f(gh)=(fg)h], itfollowsthat th