A Friendly Introduction to Abstract Algebra

404 Pages·2022·10.721 MB·English

by  Ryota Matsuura

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
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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
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and creating mathematics. i
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The author’s pedagogical philosophy is that when students abstract from a wide range n
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of examples, they are better equipped to conjecture, formalize, and prove new ideas in
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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
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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
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Readers of this text are not just consumers of mathematical knowledge. Rather, they are o c
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learning mathematics by creating mathematics. The author’s gentle, helpful writing voice i
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makes this text a particularly appealing choice for instructors and students alike. The book’s
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website has companion materials that support the active-learning approaches in the book, s
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including in-class modules designed to facilitate student exploration. u o
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For additional information
and updates on this book, visit
www.ams.org/bookpages/text-72
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TEXT/72 R
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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
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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