AMS / MAA TEXTBOOKS VOL 72 A Friendly Introduction VOL to Abstract Algebra AMS / MAA TEXTBOOKS 72 Ryota Matsuura A Friendly Introduction to Abstract Algebra offers a new approach to laying a foundation for A abstract mathematics. Prior experience with proofs is not assumed, and the book takes time to build proof-writing skills in ways that will serve students through a lifetime of learning F r and creating mathematics. i e The author’s pedagogical philosophy is that when students abstract from a wide range n d of examples, they are better equipped to conjecture, formalize, and prove new ideas in l y abstract algebra. Thus, students thoroughly explore all concepts through illuminating exam- ples before formal defi nitions are introduced. The instruction in proof writing is similarly In grounded in student exploration and experience. Throughout the book, the author carefully t r explains where the ideas in a given proof come from, along with hints and tips on how o d students can derive those proofs on their own. R y u Readers of this text are not just consumers of mathematical knowledge. Rather, they are o c t a t learning mathematics by creating mathematics. The author’s gentle, helpful writing voice i M o makes this text a particularly appealing choice for instructors and students alike. The book’s a n t website has companion materials that support the active-learning approaches in the book, s u t including in-class modules designed to facilitate student exploration. u o ra A b s t r a c t A l g e b r a For additional information and updates on this book, visit www.ams.org/bookpages/text-72 A M S / M A A P TEXT/72 R E S S 4-Color Process 402 pages on 50lb stock • Softcover • Spine 13/16” A Friendly Introduction to Abstract Algebra AMS/MAA TEXTBOOKS VOL 72 A Friendly Introduction to Abstract Algebra Ryota Matsuura MAATextbooksEditorialBoard WilliamR.Green,Co-Editor SuzanneLynneLarson,Co-Editor PaulT.Allen MarkBollman DebraS.Carney HughN.Howards WilliamJohnston EmekKose MichaelJ.McAsey ThomasC.Ratliff PamelaRichardson JeffreyL.Stuart RonTaylor RuthVanderpool ElizabethWilcox 2020MathematicsSubjectClassification.Primary12-XX,13-XX,16-XX,20-XX. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/text-72 LibraryofCongressCataloging-in-PublicationData Names:Matsuura,Ryota,1974-author. Title:Afriendlyintroductiontoabstractalgebra/RyotaMatsuura. Description: Providence, Rhode Island : American Mathematical Society, [2022] | Series: AMS/ MAAtextbooks,2577-1205;Volume72|Includesindex. Identifiers:LCCN2021062966|ISBN9781470468811(paperback)|ISBN9781470470371(ebook) Subjects:LCSH:Algebra,Abstract.|AMS:Fieldtheoryandpolynomials.|Commutativealgebra.|Associa- tiveringsandalgebras.|Grouptheoryandgeneralizations. Classification:LCCQA162.M382022|DDC512/.02–dc23/eng20220228 LCrecordavailableathttps://lccn.loc.gov/2021062966 Copyingandreprinting. Individualreadersofthispublication,andnonprofitlibrariesactingforthem, arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,providedthecustomaryac- knowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationispermit- tedonlyunderlicensefromtheAmericanMathematicalSociety.Requestsforpermissiontoreuseportions ofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. Formoreinformation,please visitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. ©2022bytheAmericanMathematicalSociety.Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. ⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 272625242322 Contents Preface xi Forthestudent xi Fortheinstructor xi Noteaboutrings xiii Roadmap xiii Acknowledgments xiv UnitI:Preliminaries 1 1 IntroductiontoProofs 3 1.1 Provinganimplication 3 1.2 Proofbycases 4 1.3 Contrapositive 6 1.4 Proofbycontradiction 7 1.5 Ifandonlyif 8 1.6 Counterexample 9 Exercises 9 2 SetsandSubsets 13 2.1 Whatisaset? 13 2.2 Setofintegersanditssubsets 14 2.3 Closure 15 2.4 Showingsetequality 17 Exercises 18 3 Divisors 21 3.1 Divisor 21 3.2 GCDtheorem 22 3.3 ProofsinvolvingtheGCDtheorem 23 Exercises 25 UnitII:ExamplesofGroups 29 4 ModularArithmetic 31 4.1 Numbersystemℤ 31 7 4.2 Equalityinℤ 32 7 4.3 Multiplicativeinverses 34 Exercises 37 v vi Contents 5 Symmetries 41 5.1 Symmetriesofasquare 41 5.2 Grouppropertiesof𝐷 44 4 5.3 Centralizer 45 Exercises 47 6 Permutations 51 6.1 Permutationsoftheset{1, 2, 3} 51 6.2 Grouppropertiesof𝑆 53 𝑛 6.3 Computationsin𝑆 54 𝑛 6.4 Associativelawin𝑆 (andin𝐷 ) 56 𝑛 𝑛 Exercises 56 7 Matrices 61 7.1 Matrixarithmetic 61 7.2 Matrixgroup𝑀(ℤ ) 62 10 7.3 Multiplicativeinverses 64 7.4 Determinant 65 Exercises 68 UnitIII:IntroductiontoGroups 71 8 IntroductiontoGroups 73 8.1 Definitionofa“group” 73 8.2 Essentialpropertiesofagroup 76 8.3 Provingthatagroupiscommutative 80 8.4 Non-associativeoperations 81 8.5 Directproduct 81 Exercises 83 9 GroupsofSmallSize 87 9.1 Smallestgroup 87 9.2 Groupswithtwoelements 88 9.3 Groupswiththreeelements 90 9.4 Sudokuproperty 91 9.5 Groupswithfourelements 92 Exercises 93 10 MatrixGroups 97 10.1 Groupsℤ and𝑈 97 10 10 10.2 Groups𝑀(ℤ )and𝐺(ℤ ) 98 10 10 10.3 Group𝑆(ℤ ) 100 10 Exercises 101 11 Subgroups 105 11.1 Examplesofsubgroups 105 11.2 Subgroupproofs 107 11.3 Centerandcentralizerrevisited 109 Exercises 110 Contents vii 12 OrderofanElement 115 12.1 Motivatingexample 115 12.2 Whendoes𝑔𝑘 =𝜀? 116 12.3 Conjugates 118 12.4 Orderinanadditivegroup 120 12.5 Elementswithinfiniteorder 121 Exercises 122 13 CyclicGroups,PartI 125 13.1 Generatorsoftheadditivegroupℤ 125 12 13.2 Generatorsofthemultiplicativegroup𝑈 127 13 13.3 Matchingℤ and𝑈 128 12 13 13.4 Takingpositiveandnegativepowersof𝑔 129 13.5 Whenthegroupoperationisaddition 131 Exercises 132 14 CyclicGroups,PartII 135 14.1 Whynegativepowersareneeded 135 14.2 Additivegroupsrevisited 136 14.3 ⟨3⟩behaves“justlike”ℤ 137 14.4 Subgroupsofcyclicgroups 138 Exercises 141 UnitIV:GroupHomomorphisms 145 15 Functions 147 15.1 Domainandcodomain 147 15.2 One-to-onefunction 148 15.3 Ontofunction 149 15.4 Whendomainandcodomainhavethesamesize 152 Exercises 153 16 Isomorphisms 157 16.1 Groupsℤ and⟨𝑔⟩: Elementsmatchup 157 12 16.2 Groupsℤ and⟨𝑔⟩: Operationsmatchup 158 12 16.3 Elementswithinfiniteorderrevisited 161 16.4 Inverseisomorphisms 162 Exercises 164 17 Homomorphisms,PartI 169 17.1 Grouphomomorphism 169 17.2 Propertiesofhomomorphisms 172 17.3 Orderofanelement 174 Exercises 175 18 Homomorphisms,PartII 179 18.1 Kernelofahomomorphism 179 18.2 Imageofahomomorphism 182 18.3 Partitioningthedomain 183 viii Contents 18.4 Findinghomomorphisms 184 Exercises 185 UnitV:QuotientGroups 189 19 IntroductiontoCosets 191 19.1 Multiplicativegroupexample 191 19.2 Additivegroupexample 193 19.3 Rightcosets 195 19.4 Propertiesofcosets 196 19.5 Whenarecosetsequal? 198 Exercises 200 20 Lagrange’sTheorem 205 20.1 MotivatingLagrange’stheorem 205 20.2 ProvingLagrange’stheorem 207 20.3 ApplicationsofLagrange’stheorem 209 Exercises 211 21 Multiplying/AddingCosets 213 21.1 Turningasetofcosetsintoagroup 213 21.2 Cosetmultiplicationshortcut 216 21.3 Cosetsof𝐻 =5ℤinℤrevisited 217 Exercises 219 22 QuotientGroupExamples 223 22.1 Quotientgroup𝑈 /𝐻revisited 223 13 22.2 Quotientgroup𝑈 /𝐻 224 37 22.3 Quotientgroup𝐺/𝐻(generalization) 225 Exercises 227 23 QuotientGroupProofs 231 23.1 Samplequotientgroupproofs 231 23.2 Collapsing𝐺into𝐺/𝐻 234 Exercises 236 24 NormalSubgroups 239 24.1 Howdoestheshortcutfailandwork? 239 24.2 Normalsubgroups: Whatandwhy 241 24.3 Examplesofnormalsubgroups 241 24.4 Normalsubgrouptest 242 Exercises 245 25 FirstIsomorphismTheorem 249 25.1 Familiarhomomorphism 249 25.2 Anotherhomomorphism 251 25.3 FirstIsomorphismTheorem 253 25.4 Findingandbuildinghomomorphisms 253 Exercises 255