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howtothinkabout abstractalgebra
HOWTOTHINKABOUT
ABSTRACTALGEBRA
laraalcock
MathematicsEducationCentre,LoughboroughUniversity
GreatClarendonStreet,Oxford, ox26dp, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
©LaraAlcock2021
Themoralrightsoftheauthorhavebeenasserted
FirstEditionpublishedin2021
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PREFACE
Thisprefaceiswrittenprimarilyformathematicslecturers,butstudentsmight finditinterestingtoo.Itdescribesdifferencesbetweenthisbookandother AbstractAlgebratextsandexplainsthereasonsforthosedifferences.
ThisbookisnotlikeotherAbstractAlgebra1 books.Itisnota textbookcontainingstandardcontent.Rather,itisdesignedas pre-readingorconcurrentreadingforanAbstractAlgebracourse. Idomeanthatitisdesignedfor reading,whichisimportantbecause studentsareoftenunaccustomedtolearningmathematicsfrombooks, andbecauseresearchshowsthatmanydonotreadeffectively.Thisbook isthereforelessdenseandmoreaccessiblethantypicalundergraduate texts.ItcontainsseriousdiscussionsofcentralAbstractAlgebraconcepts,butthesebeginwherethestudentislikelytobe.Theymake linkstoearliermathematics,refutecommonmisconceptions,andexplain howdefinitionsandtheoremscaptureintuitiveideasinmathematically sophisticatedways.ThenarrativethusunfoldsinwhatIhopeisanatural andengagingstyle,whiledevelopingtherigourappropriateforundergraduatestudy.
Becauseofthisaim,thebookisstructureddifferentlyfromothertexts. Part1containsfourchaptersthatdiscussnotthecontentofAbstract Algebrabutitsstructure,explainingwhatitmeanstohaveacoherent mathematicaltheoryandwhatittakestounderstandone.Thereisno ‘preliminaries’chapter;instead,notationsanddefinitionsareintroduced wheretheyarefirstneeded,meaningthattheyarespreadacrossthetext (thoughasymbollistisprovidedonpagesxiii–xiv).Thismeansthata
1 ‘AbstractAlgebra’shouldprobablynothaveupper-case‘A’s,butIwanttomakethe subjectnamedistinctfromotherusesofrelatedterms.
studentreadingforreviewmightneedtousetheindexmorethanusual, sotheindexisextensive.
Aseconddifferenceisthatnotallcontentiscoveredatthesamedepth. ThefivemainchaptersinPart2eachcontainextensivetreatmentoftheir centraldefinition(s),especiallywherestudentsareknowntostruggle. Theyalsodiscussselectedtheoremsandproofs,someofwhichareused tohighlightstrategiesandskillsthatmightbeusefulelsewhere,andsome ofwhichareusedtodrawoutstructuralsimilaritiesandexplaintheory development.Butthesechaptersaimtoprepareastudenttolearnfroma standardAbstractAlgebracourse,ratherthantocoveritsentirecontent.
Athirddifferenceisthattheorderofthecontentisrelativelyunconstrainedbylogicaltheorydevelopment.Theoryisexplicitlydiscussed: Part1providesinformationontherolesofaxioms,definitions,theorems andproofs,andPart2encouragesattentiontologicalargument.But numeroussectionsprovideexamplesbeforeinvitinggeneralization,or introducetechnicaltermsinformallybeforetheyaredefined,orobserve phenomenatobeformalizedlater.Itakethisapproachwithcare,highlightinginformalityandnotingwhereformalversionscanbefound.But IconsideritusefulbecausethecentralideasofearlyAbstractAlgebraare sotightlyinterwoven—reallyIwouldliketointroduceallofthebook’s mainideassimultaneously.Obviouslythatisimpossible,butIdowant tokeepsomepaceinthenarrative,toprioritizeconceptsandimportant relationshipsovertechnicalities.Irealize,ofcourse,thatthisapproach goesagainstthegrainofmathematicalpresentationandmeansthatthisis notabookthatatypicalcoursecould‘follow’.ButIamcontentwiththat— theworldisfullofstandardtextbooksdevelopingtheoryfromthebottom up,andmyaimistoprovideanalternativewithafocusonconceptual understanding.
Finally,thisbookexplicitlydiscusseshowstudentsmightmakesense ofAbstractAlgebraasitispresentedinlecturesandinotherbooks. Irealizethatthis,too,iscontentious:manymathematicslecturersplace highvalueonconstructingideasandarguments,andsomehaveworked hardtodevelopinquiry-basedAbstractAlgebracourses.Iamallfor inquiry-basedcourses,andforanywell-thought-outapproachthatallows studentstoreinventmathematicalideasthroughindependentorcollaborativeproblemsolving.Buttherealityisthatmostmathematicslectures arestilljustthat:lectures.Manylecturersareconstrainedbyclasssizes
wellintothehundreds,andflippedclassroommodelsmightpromote studentengagementbutitisnotobvioushowtousethemeffectivelyto developthetheoryofAbstractAlgebra.Becauseofthis,andbecausefew studentsfolloweverydetailoftheirlectures,animportantstudenttask istomakesenseofwrittenmathematics.Researchshowsthatthetypical studentiscapableofthisbutill-informedregardinghowtogoaboutit. Thisbooktacklesthatproblem—itaimstodeliverstudentswhodonot yetknowmuchAbstractAlgebrabutwhoarereadytolearn.
Abooklikethiswouldbeimpossiblewithoutworkbynumerous researchersinmathematicseducationandpsychology.Inparticular,the self-explanationtraininginChapter3wasdevelopedincollaboration withMarkHoddsandMatthewInglis(seeHodds,Alcock&Inglis,2014) onthebasisofearlierresearchonacademicreadingbyauthorsincluding AinsworthandBurcham(2007),Bielaczyc,Pirolli,andBrown(1995), andChi,deLeeuw,ChiuandLaVancher(1994).Moreinformationon thestudiesweconductedcanbefoundinAlcock,Hodds,RoyandInglis (2015),anarticleinthe NoticesoftheAmericanMathematicalSociety. Thebibliographycontainsextensivereferencestodecades’worthof researchonstudentlearningaboutspecificconceptsinAbstractAlgebra andaboutgeneralproof-basedmathematics.Iencourageinterested readerstoinvestigatefurther.
MyspecificthanksgotoAntEdwards,TimFukawa-Connelly,Kevin Houston,ArtiePrendergast-Smith,AdrianSimpson,KeithWeber,and IroXenidou-Dervou,allofwhomgavevaluablefeedbackonchapter drafts.Similarly,tocarefulreadersRomainLambert,NeilPratt,and SimonGoss,whowerekindenoughtopointouterrors.Iamparticularly indebtedtoColinFoster,whoreadadraftoftheentirebookbeforeits chapterswenttoanyoneelse.Iamalsogratefulasevertotheteam atOxfordUniversityPress,includingDanTaber,KatherineWard, ChandrakalaChandrasekaranandRichardHutchinson.Finally,this bookisdedicatedtoKristianAlcock,whohasneverknownmenottobe writingit.Ithinkhewillbegladandamazedtoseeitinprint.
Part2TopicsinAbstractAlgebra
5BinaryOperations
5.1Whatisabinaryoperation?
5.4Binaryoperationsonfunctions
5.5Matricesandtransformations
5.6Symmetriesandpermutations
5.7Binaryoperationsasfunctions
6GroupsandSubgroups
6.1Whatisagroup?
6.2Whatisasubgroup?
6.3Cyclicgroupsandsubgroups
SYMBOLS
SYMBOLS
INTRODUCTION
ThisshortintroductiondiscussestheplaceofAbstractAlgebraintypical undergraduatedegreeprogrammesandthechallengesitpresents.Itthen explainsthisbook’scontentandintent.
MathematicsstudentstypicallyencounterAbstractAlgebraas oneoftheirfirsttheorems-and-proofscourses.Forthosein UK-likesystems,wherepeoplespecializeearly,itmightbe taughtatthebeginningofamathematicsdegree,orperhapsinthesecond termorsecondyear.ForthoseinUS-likesystems,wherepeoplespecialize late,itwillmorelikelybeanupper-levelcourseformathematicsmajors intheirjuniororsenioryear.Eitherway,acoursemightnotactuallybe called AbstractAlgebra.Thatnameisanumbrellatermforthetheoryof groups,rings,fieldsandrelatedstructures,soafirstcoursemightbetitled GroupTheory or Groups,RingsandFields orsomethingmorebasiclike SetsandGroups.
Whatevertheprecisearrangements,AbstractAlgebraisusuallystudied inparallelwithothersubjectssuchasAnalysis.BothAbstractAlgebraand Analysisinvolveashiftinmathematicalemphasis:studentswhothinkof mathematicsasasetofalgorithmsmustnowlearntofocusondefinitions, theoremsandproofs.Part1ofthisbookprovidesadviceonthat.But thetwosubjectscanfeelquitedifferent,andawarenessofthedifferences mighthelpwithunderstandingtheirrespectivechallenges.
InAnalysis,themainchallengeisthereasoning:thesubjectmakes heavyuseoflogicallycomplexstatements,whichfewstudentsarewell equippedtoprocess.Butitsobjectsarerelativelygraspable:numbers, sequences,seriesandfunctionsarealreadyfamiliarorreadilyrepresentedindiagrams.InAbstractAlgebra,themainchallengeisalmost theopposite.Thelogicismorestraightforward,butmanyoftheobjects arelessfamiliarandlesseasytorepresent.Forstudents,thiscanrender
thesubjectsomewhatmeaningless—eventhosewhodowellmightnot developastrongsenseofwhatitis‘about’.Thatwasmyexperience. Despitehavingaverygoodlecturer,1 IneverreallylikedAbstractAlgebra becauseIneverreally got it.Icouldperformthemanipulations,applythe theorems,andreconstructtheproofs,butIdidn’treallyunderstandwhat itallmeant.
Inowbelievethatthishappenedfortworeasons.Thefirstisgeneral butparticularlyrelevantinAbstractAlgebra:mathematicsishierarchical, witheachlevelbuildingonthelast.Shiftingupaleveloftenrequires compressingsomeaspectofyourunderstandinginordertotothink ofitinrelationtofourorfivenewthings.Ifyouhaven’tcompressed itenough,thisisdifficult,andhigherlevelscanseemlikemeaningless symbol-pushing.AbstractAlgebrainvolvesalotofcompression,andthis bookwillpointoutexplicitlywhereitisneeded.
ThesecondreasonisthatIdidn’taccessgoodrepresentationsfor AbstractAlgebra’skeyideas.Ilikevisualrepresentations—imagesthat enablemeto‘see’howconceptsarerelatedandtodevelopintuitionfor whythingsworkastheydo.Idon’toftengetthatfeelingfromalgebraic arguments,nomatterhowsureIamthateachstepisvalid.AndAbstract Algebradidn’tseemtohavemanyvisualrepresentations,soIdidn’tfind muchtoholdonto.Idon’tblamemylecturerforthis—Ifailedrather badlytokeepup,soIdidn’tfollowhislectureseffectivelyandIprobably missedsomeenlighteningexplanations.Butintuitionforthesubjectcan bedevelopedusingdiagramsandtables,sothisbookcontainsmany ofthose.
ThisbookalsocontainsexplicitdiscussionofwhatAbstractAlgebra is,beginninginChapter1withsectionsonwhatisabstractabout AbstractAlgebra,andwhatisalgebraicaboutit.Theremainderof Part1discussesaxiomsanddefinitionsandtheirrolesinmathematical theory(Chapter2),theoremsandproofsandproductivewaystointeract withthese(Chapter3),andresearch-basedstrategiesforeffective learning(Chapter4).PleasereadChapter4evenifyouareasuccessful
1 IntheUKwesay‘lecturer’wherethoseinUS-likesystemsmightsay‘instructor’ or‘professor’.
student—youmightfindthatyoucantweakyourstrategiestoimprove yourlearningorreduceyourworkloadorboth.2
Part2coverstopicsinAbstractAlgebra,startingwithbinaryoperations (Chapter5)andmovingontogroupsandsubgroups(Chapter6), quotientgroups(Chapter7),isomorphismsandhomomorphisms (Chapter8)andrings(Chapter9).Becausethisisnotastandard textbook,itdoesnottryto‘cover’alloftherelevantcontentforthese topics.Instead,ittreatsthemainideasindepth,usingexamplesand visualrepresentationstoexplainwaystothinkaboutthemaccurately. Eachchapteralsoincludesselectedtheoremsandproofsanddiscusses relationshipsbetweentopics.
BecauseAbstractAlgebraisatightlyinterconnectedtheory,early chaptersoftentouchonideasnotformalizeduntillater.Forthisreason, Irecommendreadingthewholebookinorder,althougheachchapter shouldalsobereadableasaself-containedunit.Ifyougetstuck,rememberthatthereisanextensiveindex,andthatpagesxiii–xivlistwhere symbolsareexplainedinthetext.Ifitispractical,Ialsorecommend readingtheentirebookbeforestartinganAbstractAlgebracourse.I intendtosetyouupwithmeaningfulunderstandingofthemainideasand agoodgraspofhowtolearneffectively,soyouwilllikelygetmaximum benefitbyreadingbeforeyoustart.However,ifyouhavecometothis bookbecauseyourcoursehasbegunandyoufindyourselflostinthe abstractions,itshouldprovideopportunitiestoreworkyourunderstandingsothatyoucanengageeffectively.Eitherway,IhopethatAbstract Algebradeepensyourunderstandingofbothfamiliarmathematicsand higher-leveltheory.
2 Comparedwith HowtoStudyforaMathematicsDegree anditsAmericancounterpart HowtoStudyasaMathematicsMajor,theadviceinChapter4iscondensed, specifictoAbstractAlgebra,andmoreexplicitlylinkedtoresearchonlearning. INTRODUCTION xvii
PART1 StudyingAbstractAlgebra
WhatisAbstractAlgebra?
ThischaptercontrastsAbstractAlgebrawiththealgebrastudiedinearlier mathematics.Ithighlightsthesubject’sfocusonvalidityofalgebraicmanipulationsacrossarangeofstructures.Itthendescribesthreeapproachescommon inAbstractAlgebracourses:aformalapproach,anequation-solvingapproach andageometricapproach.
1.1WhatisabstractaboutAbstractAlgebra?
AbstractAlgebraisabstractinthesamesenseinwhichother humanthinkingisabstract:itsconceptscanbeinstantiatedin multipleways.Forinstance,yourecognizethingsliketreesand windows.Youcandothatbecauseyouunderstandtheabstractideas‘tree’ and‘window’,andyoucanmatchthemtoobjectsintheworld.Youdonot needtolookatonetreetoidentifyanother;youdoitbyreferencetothe abstractidea.
Now,treesandwindowsarephysicalobjects—youcanwalkupand touchthem,andidentifythembysight.Butyoucanthinkaboutconcepts thataremoreabstract,too.Forinstance,youcanidentifyanaunt.You dothatbyreferencetoacriterion:isthisafemalepersonwithasibling whohaschildren?Ifyes,it’sanaunt.Ifno,it’snot.Moreover,youcan thinkaboutabstractconceptsthatarenotsingleobjects,like family. Afamilyincludesmultiplepeople—perhapsmany—whoarerelatedto oneanother—geneticallyorbymarriageorbyothercaringrelationships. Familiesvaryalot,andyoucouldn’tnecessarilyrecognizeafamilyby
sightorbycheckingsimplecriteria.Butyouneverthelessunderstand theidea.
AbstractAlgebraisaboutconceptsthataresomewhatlikeeachof thesemoreabstractideas.Theyarelikeauntsinthattheyaredefinedby criteria.AbstractAlgebraisstricter,though.Everydayhumanconcepts, evendefinedoneslike‘aunt’,tendtobeusedflexibly.Isyourmum’s brother’sfemalepartneryouraunt?Maybe,maybenot.AndwhereIgrew up,adultfemaleneighboursandfriendswerecommonlyreferredtoas ‘Auntie’,evenwheretherewerenofamilyrelationships.Suchflexibility doesn’thappeninAbstractAlgebra,becausemathematicalconceptsare specifiedbyprecisedefinitionsaboutwhichallmathematiciansagree.1
TheconceptsofAbstractAlgebraarelikefamiliesinthatnotallare justsinglethings:somearesetswithparticularinternalrelationships.This meansthattheycanbebigandcomplex,thoughthatdoesnotnecessarily makethemhardtothinkabout.Asingletree,afterall,mighthavetens ofbranchesandthousandsofleaves—ithaslotsofinternalstructure, butyoucantreatitasasinglething.Mathematicalobjectscanbelike thistoo.Forinstance, thesetofallevennumbers isinfiniteandhaslots ofinternalstructure,butagainyoucantreatitasasinglething.Such thinkingisimportantinAbstractAlgebra:oftenitisusefultoswitch betweenexamininganobject’sinternalstructureandthinkingofitasa unifiedwhole.
1.2WhatisalgebraicaboutAbstractAlgebra?
TounderstandwhatisalgebraicaboutAbstractAlgebra,itisprobably usefultoconsiderwhatisalgebraicaboutearlieralgebra.Manystudents thinkofalgebraassomethingyou do,where doingalgebra meansmanipulatinganexpressionorequationinvalidwaystoarriveatanother.Thisis ofteninserviceofagoal:solvingamechanicsproblem,say.Andthinking ofalgebrainthiswayisnotwrong—certainlyitcapturesmoststudents’
1 Moreaccurately,mathematiciansagreeabouttheprinciplethatconceptsshould bedefinedinthisway,andmostundergraduatemathematicsworkslikethis.But, historically,thereweredebatesabouthowbesttodefineeverything,andsuchdebates continueindevelopingsubjects.
4 WHATISABSTRACTALGEBRA?
experiencepriortoundergraduatemathematics.Butitisnotenoughto grasptheaimsofAbstractAlgebra.
AbstractAlgebrafocusesnotonperformingalgebraicmanipulations butonunderstandingthemathematicalstructuresthatmakethose manipulationsvalid.ToseewhatImean,considerthisalgebraicargument (thearrow‘⇒’means‘implies’).
Probablyyoucanwritesuchargumentsquickly,fluentlyandwithonly occasionalerrors.Butwhyexactlyiseachstepvalid?Onestep(which?) assumesthat xy = yx.Thisisvalidbecausemultiplicationis commutative,meaningthat xy and yx alwaystakethesamevalue.Anotherstep (which?)assumesthatif x2 = 0then x = 0.Thisisvalidbecause0isthe onlynumberthat,whensquared,gives0.Butsuchassumptionsrelyon propertiesofoperationsandobjects.Multiplicationiscommutative,but notalloperationssharethisproperty.Divisionisnotcommutative,for instance: x/y couldnotbereplacedby y/x.Andnotallobjectsbehave likenumbers.If x and y were2 × 2matrices,2 wecouldnotassumethat xy = yx becausematrixmultiplicationworkslikethis:
So,forexample, (12 34)(56 78) = ( 5 + 146 + 16 15 + 2818 + 32) = (1922 4350)
2 Thisbookincludesexamplesbasedonmatricesandcomplexnumbers.Ifyou arestudyinginaUK-likesystemandhavenotcomeacrossthese,youcanfind introductionsinA-levelFurtherMathematicstextbooksorreliableonlineresources.
but (56 78)(12 34) = (5 + 1810 + 24 7 + 2414 + 32) = (2334 3146).
Similarly,wecouldnotassumethatif x2 = 0then x = 0.Thematrix (01 00)
isnotthezeromatrix,butnevertheless (01 00)(01 00) = (0 + 00 + 0 0 + 00 + 0) = (00 00).
Algebraicvaliditythereforedependsuponpropertiesofboth binary operations,includingmultiplication,divisionandotherstobediscussed inChapter5,andthe sets onwhichtheseoperate,whichmightbesetsof numbers,matricesorobjectsofothertypes—again,seeChapter5.Binary operationsmightbecommutativeonsomesetsbutnotonothers.Insome setstherearefewwaystocombineobjectstogivezero;inothers,there aremany.Thus,‘facts’thataretrueforoneoperationononesetdonot necessarilyholdelsewhere,andAbstractAlgebrarequiresconcentration toensurethatyoudonotovergeneralizefromafamiliarcontext.Iwon’t lie:thisishard.Whenyouareaccustomedto‘doing’algebrainnumerical contexts,itmightnotrequiremucheffort.Themanipulationsbecome naturalenoughthatyoudothemeasily,muchasyoumightwalkortype easily.Andconcentratingonsomethingthatyoudoeasilyfeelsweirdand disruptive.Ifyouconcentrateonyourmusclesasyouwalk,youbecome slowandungainly.Ifyoucantouch-type,butyouforceyourselftolookat thekeyboardandthinkaboutwhichlettersyouwant,youmightfindthat youcan’ttypeatall.Focusingonwhyalgebraicmanipulationsarevalid canfeelsimilar:slow,clunkyandthereforelikeastepbackwardinyour learningratherthanastepforward.Ittakesdiscipline,andforawhile mightfeelfrustrating.
Buttherewardisworthit.Differencesoccurinthedetail,butat largerscalesAbstractAlgebrarevealsstrikinglysimilarstructures.Sets
andoperationsthatappearquitedifferentturnouttohavenumerous commonproperties.Thissortofthingexcitespuremathematicians.But evenifyou’renotbynatureoneofthose—ifyou’reintomathematicsfor itspracticalapplications—Iencourageyoutobeopentotheideas.There ispleasureinrecognizingstructuresthatcutacrossthesubject.
1.3ApproachestoAbstractAlgebra
ToappreciateAbstractAlgebra,youwillneedtoengageeffectivelywith yourcourse.Andcoursesdiffer.Lecturershaveindividualapproaches, andeventhosewhofollowbookswillemphasizesomethingsandskip overothers.Thatsaid,manyAbstractAlgebracoursesstartwithgroup theory.Somestartwithringsoracombinationofstructuresbut,asthe grouptheorybeginningiscommon,itformsthegreaterpartofthisbook.
Inteachinggrouptheory,therearethreebroadapproaches:a formal approach,an equation-solving approachanda geometric approach.Your coursewilllikelyhavesomethingincommonwithatleastoneofthese, andthisbookwilldrawonallthree.HereIwilldescribeeachinturn, commentingontheiradvantagesanddisadvantages.Thedescriptions arenecessarilycaricatures,buttheygiveaflavourofwhatyoumight encounter.
A formal approachisinonesensethesimplest.Lecturerstakingthis approachtendtobeginwithdefinitions,likethatforgroupshownbelow.3 ThisisexplainedindetailinChapters2,5and6;fornow,youmightlike toknowthatthesymbol‘∈’means‘(which)isanelementof’andisoften readsimplyas‘in’.
Definition: A group isaset G withabinaryoperation ∗ suchthat:
Identity thereexists e ∈ G suchthatforevery g ∈ G,
Inverses forevery g ∈ G, ∃g′ ∈ G suchthat g ∗ g′ = g′ ∗ g = e.
3 Ifyourcourseusesadefinitionthatomitsthe closure criterion,seeSection5.7.
;
