A course in abstract algebra

493 Pages·2016·10.278 MB·English

by  Nicholas Jackson

N I C H O L A S JAC K S O N
A C O U R S E I N
A B S T R A C T A L G E B R A
DRAFT: JUNE 17, 2016

To Abigail and Emilie

We may always depend upon it that
algebra which cannot be translated
intogoodEnglishandsoundcommon
senseisbadalgebra.
—WilliamKingdonClifford
(1845–1879),
TheCommonSenseoftheExactSciences
(1886)21
Preface
Abstractalgebraisafascinating,versatileandpowerfulsubject
withmanyimportantapplicationsnotjustthroughoutthewider
world of mathematics, but also in several other disciplines as well.
Grouptheory,thefocusofthefirstsevenchaptersofthisbook,ispar-
ticularlygoodatdescribingsymmetry. Ittherebyenablesmolecular
chemistsandcrystallographerstostudyandunderstandthestructure
andpropertiesofmoleculesandcrystals,anditgivesparticlephysi-
cistsvaluableinsightsintothefundamentalparticlesandforcesthat
formouruniverse. Finitefieldsandsporadicgroupshaveimportant
applicationsincryptographyandinformationtheory,intheconstruc-
tionofstrongpublic-keycryptosystemsanderror-correctingcodesfor
secureandreliabletransmissionorstorageofdata.
Butitsabstractnature,thethingthatmakesitsousefulandversatile,
canalsosometimesrenderitopaque,dryanddifficulttounderstand
forstudentsmeetingthesubjectforthefirsttime. Istruggledconsider-
ablywithmuchofthismaterialwhenIwasastudent,andtherewere
certainlyseveraloccasionswhenIlostsightofwhywewerelearning
aparticulartopicandhowitjoinedupwitheverythingelse. Theeffort
ultimatelypaidoffbutIwasleftwonderingwhetheritneededtobe
sodifficult.
In some sense, the answer is yes: mathematics requires a level of
formality and rigour greater than probably any other subject, and
soonerorlaterwemustputintheworkneededtofullyunderstand
andinternalisetheconceptsandtheoremsunderdiscussion. Toborrow
Euclid’sfamousattributedcommenttotheEgyptianpharaohPtolemy
ISoter(c.367–c.283BC),thereisnoroyalroadtoalgebra. ButIremain
convincedthatthejourneycanatleastbedemocratisedtosomeextent.
IntheintroductiontoHermanHesse’snovelTheGlassBeadGame,the
narrator remarks that no textbook of the Game will ever be written,
becausenobodywhohasdevotedthelongyearsofstudynecessaryto
masteritwouldhaveanyinterestinmakingiteasierforanyoneelse
doso. ThereweretimeswhenIalmostwonderedifsomethingsimilar
vi a course in abstract algebra
wasoccasionallythecasewithmathematics.
Butinmyfinalyearasanundergraduate,tryingtounderstandtensor
calculusandgeneralrelativity,IfoundRayd’Inverno’sexcellenttext-
1RA d’Inverno, Introducing Einstein’s bookIntroducingEinstein’sRelativity,1 whichconvincedmethatitwas
Relativity, Clarendon Press, Oxford indeedpossibletoexplainadifficultsubjectinalucid,engagingand
(1992).
readablemanner.
Towardstheendofanotherwisepositivebookreview,thealgebraic
topologistFrankAdams(1930–1989)eloquently(andconstructively)
2JF Adams, Review of ‘Algebraic K– lamentstheshortcomingsofmanyalgebratextbooks:2
Theory’,BulletinoftheLondonMathe-
maticalSociety2.2(1970)233–238. However,Idofeelimpelledtotrytosaywhatneedstobesaidabout
a whole way of writing books on algebra, from Van der Waerden to
CartanandEilenberg. Look,youpeoplearewritingaboutasubjectI
love. IwouldechoHardyandsay“thesubjectissoattractivethatonly
3GHHardyandEMWright,AnIntro- extravagantincompetencecouldmakeitdull”.3 Allthesame,wehave
ductiontotheTheoryofNumbers,Claren- bookswhichdothat. Theyareaustereandtheyarearid. Whyisthis?
don Press, Oxford (1938), preface to
firstedition.
Hediscussespossiblereasonswhysomemathematicstextbooksare
difficult to read, suggesting in particular that the author is often so
convincedofthefascinationoftheirsubjectthattheytakeitforgranted
thattheirreaderswillalsoseeitwithoutfurtherprompting.
4RRSkemp,ThePsychologyofLearning Early in his book on the psychology of learning mathematics,4 the
Mathematics,secondedition,Penguin mathematicaleducationalistRichardSkemp(1919–1995)proposestwo
(1986),pages30–33.
basic principles of mathematics education: that mathematical con-
ceptscanonlybeproperlycommunicatedwithillustrativeexamples,
andthatmathematicalconceptsbuildanddependoneachothertoa
greaterextentthaninalmostanyothersubject. Animperfectunder-
standingofoneconceptcanendangerthosethatfollowfromit.
In this book I have done my best to present a comprehensive and
readable treatment of all the topics one might reasonably expect to
meet in a standard undergraduate algebra course. Bearing in mind
both of Skemp’s main principles, I’ve tried to introduce most new
definitionsandtheoremswithoneormoreillustrativeandmotivating
examples, and to link them together into a coherent narrative. I’ve
also included some biographical notes to put the subject into some
sort of historical context and also in the hope that readers will find
thismaterialinteresting.
Each chapter ends with a short, cross-referenced summary which I
hope will be useful for revision purposes. In addition, I’ve drawn
inspiration(thatis,shamelesslystolen)fromthelayoutofd’Inverno’s
bookandthoseofthestatisticianandinformationdesignerEdward
Tufte.
I have also included some additional topics that will hopefully be
vii
usefulforthosewishingtostudythesubjecttoamoreadvancedlevel,
although there are a number of omissions, and in some cases this
additionalmaterialmostlyjustpointstheway.
The main aim of this book is to provide a clear, detailed and com-
prehensiveaccountofthetopicscoveredinatypicalundergraduate
courseonabstractalgebra. IhopethatI’vemostlysucceeded,butif
not,thentoechoHardyandWrightIhopethatmyincompetenceat
leasthasn’tbeentooextravagant.
Exercises On two occasions I have been asked,
–“Pray,Mr. Babbage,ifyouputinto
the machine wrong figures, will the
Serving suggestions rightanswerscomeout?”Inonecasea
memberoftheUpper,andintheother
amemberoftheLower,Houseputthis
Acknowledgements question. Iamnotablerightlytoap-
prehendthekindofconfusionofideas
thatcouldprovokesuchaquestion.
IwouldliketothankKeithMansfield,ClareCharles,VikiMortimer
—CharlesBabbage(1791–1871),
andDanielTaberofOxfordUniversityPress,foralltheirhelp,advice PassagesfromtheLifeofaPhilosopher
andpatience. (1864)67
InthisbookIhaveattemptedtogive,
Numerous friends and colleagues offered advice, or read and com- under the chapters devoted to cook-
ery,anintelligiblearrangementtoev-
mentedondraftsofthisbook,andIthankthemallfortheirgenerosity
ery recipe, a list of the ingredients, a
andmanyhelpfulsuggestions,whileatthesametimeacceptingsole plainstatementofthemodeofprepar-
responsibilityforanyremainingerrors. ApartiallistincludesAndrew ingeachdish,andacarefulestimateof
itscost,thenumberofpeopleforwhom
Brendon-Penn,GavinBrown,InnaCapdebosq,MarkCummings,Sîan
itissufficient,andthetimewhenitis
Fryer,JamesGrime,DerekHolt,HeatherMcCluskey,NickMills,Colin seasonable.
Rourke, Jochen Voss, Charles Walkden, Bruce Westbury and Colin —IsabellaBeeton(1836–1865),
Beeton’sBookofHouseholdManagement
Wright,andIapologisetoanyoneI’veinadvertentlymissed. (1861)iii
Finally,andmostofall,IthankmywifeAbigailDaviesforherencour- Bernard of Chartres used to say that
we were as dwarves standing on the
agementandsupport,andourdaughterEmilie,whowasbornduring
shouldersofgiants,sothatwecould
the writing of this book, and without whom I might have finished seemorethanthem,andfurther;not
becausewehavesharpersightorare
slightlysooner,butwhohasmadeitallseemmuchmoreworthwhile.
taller,butbecauseweareraisedupand
heldaloftbytheirgreatheight.
—JohnofSalisbury(c.1120–1180),
Credits MetalogiconIII:4
Youhaveaddedmuchseveralways,&
especiallyintakingthecoloursofthin
NicholasJackson,
platesintophilosophicalconsideration.
Coventry,September2015 IfIhaveseenfurtheritisbystanding
onthesholdersofGiants.
—SirIsaacNewton(1642–1727),
LettertoRobertHooke(1635–1703),
dated5February1676

Contents
Preface v
Exercises vii
Servingsuggestions vii
Acknowledgements vii
Credits vii
1 Groups 1
1.1 Numbers 1
1.2 Matrices 17
1.3 Symmetries 20
1.4 Permutations 24
2 Subgroups 39
2.1 Groupswithingroups 39
2.2 CosetsandLagrange’sTheorem 49
2.3 Euler’sTheoremandFermat’sLittleTheorem 62
3 Normal subgroups 69
3.1 Cosetsandconjugacyclasses 69
3.2 Quotientgroups 80
3.A Simplegroups 91
4 Homomorphisms 103
4.1 Structure-preservingmaps 103
4.2 Kernelsandimages 110
4.3 TheIsomorphismTheorems 116
5 Presentations 133
5.1 Freegroups 134
5.2 Generatorsandrelations 144
5.3 Finitelygeneratedabeliangroups 165
x a course in abstract algebra
5.A Cosetenumeration 178
5.B Transversals 186
5.C Triangles,braidsandreflections 198
6 Actions 205
6.1 Symmetriesandtransformations 205
6.2 Orbitsandstabilisers 213
6.3 Counting 220
7 Finite groups 233
7.1 Sylow’sTheorems 233
7.2 Seriesofsubgroups 244
7.3 Solubleandnilpotentgroups 255
7.4 Semidirectproducts 272
7.5 Extensions 277
7.A Classificationofsmallfinitegroups 291
8 Rings 317
8.1 Numbers 317
8.2 Matrices 326
8.3 Polynomials 328
8.4 Fields 332
8.A Modulesandrepresentations 338
9 Ideals 343
9.1 Subrings 343
9.2 Homomorphismsandideals 351
9.3 Quotientrings 359
9.4 Primeidealsandmaximalideals 368
10 Domains 377
10.1Euclideandomains 377
10.2Divisors,primesandirreducibleelements 385
10.3Principalidealdomains 391
10.4Uniquefactorisationdomains 395
10.AQuadraticintegerrings 408
11 Polynomials 421
11.1Irreduciblepolynomials 421
11.2Fieldextensions 425