A First Course in Group Theory

300 Pages·2021·3.885 MB·English

by  Bijan Davvaz

Bijan Davvaz
A First Course
in Group
Theory
A First Course in Group Theory
Bijan Davvaz
A First Course in Group
Theory
BijanDavvaz
DepartmentofMathematics
YazdUniversity
Yazd,Iran
ISBN978-981-16-6364-2 ISBN978-981-16-6365-9 (eBook)
https://doi.org/10.1007/978-981-16-6365-9
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Preface
Theaimofthisbookistoprovideareadableaccountoftheexamplesandfundamental
resultsofgroupsfromatheoreticalpointofviewandageometricalpointofview.
The concept of a group is one of the most fundamental in modern mathematics.
Groupsaresystemsconsistingofasetofelementsandabinaryoperationthatcan
beappliedtotwoelementsoftheset,whichtogethersatisfycertainaxioms.These
require that the group is closed under the operation (the combination of any two
elements produces another element of the group), that it obey the associative law,
thatitcontainsanidentityelement(which,combinedwithanyotherelement,leaves
thelatterunchanged),andthateachelementhaveaninverse(whichcombineswith
anelementtoproducetheidentityelement).
Sincethebookdoesnotmakeanyassumptionsaboutthereader’sbackground,it
issuitablefornewcomerstogrouptheoryandeventhosewhoneverstudiedalgebra.
Toexplainmanysubjectsweusedfiguresandimagestohelpthereaders.
Thebookisorganizedintoelevenchapters.Togettoanydepthingrouptheory
requiressettheory,combinatorics,numbertheory,matrixtheory,andgeometry.Thus,
not only we have devoted Chaps. 1–2 to the introductory concepts of set theory,
combinations,numbertheory,andsymmetry,butwehavealsodevotedsomeparts
ofChap.7tomatrixtheory.Althoughthemainsubjectofthebookisgroupsbutin
manypartswewillneedsomeinformationaboutotheralgebraicstructures,likerings,
fieldsandvectorspaces.InChap.3,wegivethedefinitionsofgroupandsubgroup,
examplesandsomeelementaryproperties.Severalexcellentandimportantexamples
of groups like cyclic groups, permutation groups, group of arithmetical functions,
matrixgroupsandlineargroupsareinvestigatedinChaps.4–7.InChap.8,Lagrange’s
theoremasthemostimportanttheoreminfinitegroupsisdiscussed.Chapters9–11
address the normal subgroups, factor groups, derived subgroup, homomorphism,
isomorphismandautomorphism ofgroups.Aconsequence isCayley’s theorem in
whicheverygroupcouldberealizedasapermutationgroup.
Somechaptersorsectionsarelabelledasoptional;thismeansthatthereaderscan
ignore them for the first study. Each section ends in a collection of exercises. The
purposeoftheseexercisesistoallowstudentstotesttheirassimilationofthematerial,
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to challenge their knowledge and ability. Moreover, at the end of each chapter we
havetwospecialsections:(1)Worked-OutProblems;(2)SupplementaryExercises.
In each worked-out problems section, I decided to try teach by examples, by
writingouttosolutionstoproblems.Inchoosingproblems,threemajorcriteriahave
beenconsidered,tobechallenging,interestingandeducational.Moreover,thereare
many exercises at the end of each chapter as supplementary exercises. They are
harderthanbeforeandservetopresenttheinterestingconceptsandtheoremswhich
arenotdiscussedinthetext.
Thelistofreferencesattheendofthebookisconfinedtoworksactuallyusedin
thetext.
Yazd,Iran BijanDavvaz
Contents
1 PreliminariesNotions .......................................... 1
1.1 SetsandEquivalenceRelations ............................. 1
1.2 Functions ............................................... 8
1.3 OrderedSets ............................................. 11
1.4 CombinatorialAnalysis ................................... 14
1.5 DivisibilityandPrimeNumbers ............................ 18
1.6 Worked-OutProblems .................................... 25
1.7 SupplementaryExercises .................................. 28
2 SymmetriesofShapes ......................................... 31
2.1 Symmetry ............................................... 31
2.2 Translations ............................................. 34
2.3 RotationSymmetries ..................................... 36
2.4 MirrorReflectionSymmetries .............................. 38
2.5 CongruenceTransformations ............................... 41
2.6 Worked-OutProblems .................................... 43
2.7 SupplementaryExercises .................................. 45
3 Groups ....................................................... 47
3.1 AShortHistoryofGroupTheory ........................... 47
3.2 BinaryOperations ........................................ 49
3.3 SemigroupsandMonoids(Optional) ........................ 53
3.4 GroupsandExamples ..................................... 59
3.5 TurningGroupsintoLatinSquares(Optional) ................ 72
3.6 Subgroups ............................................... 76
3.7 Worked-OutProblems .................................... 81
3.8 SupplementaryExercises .................................. 85
4 CyclicGroups ................................................ 87
4.1 GroupofIntegersModulon ............................... 87
4.2 CyclicGroups ........................................... 91
4.3 GeneratingSets .......................................... 99
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4.4 Worked-OutProblems .................................... 101
4.5 SupplementaryExercises .................................. 102
5 PermutationGroups ........................................... 105
5.1 InverseFunctionsandPermutations ......................... 105
5.2 SymmetricGroups ....................................... 108
5.3 AlternatingGroups ....................................... 119
5.4 Worked-OutProblems .................................... 124
5.5 SupplementaryExercises .................................. 127
6 GroupofArithmeticalFunctions(Optional) ..................... 131
6.1 ArithmeticalFunctions .................................... 131
6.2 DirichletProductandItsProperties ......................... 136
6.3 MultiplicativeFunctions ................................... 139
6.4 Worked-OutProblems .................................... 143
6.5 SupplementaryExercises .................................. 147
7 MatrixGroups ................................................ 149
7.1 IntroductiontoMatrixGroups .............................. 149
7.2 MoreAboutVectorsinRn ................................. 166
7.3 RotationGroups ......................................... 170
7.4 ReflectionsinR2 andR3 .................................. 175
7.5 TranslationandScalingMatrices ........................... 180
7.6 DihedralGroups ......................................... 183
7.7 QuaternionGroup ........................................ 189
7.8 Worked-OutProblems .................................... 191
7.9 SupplementaryExercises .................................. 193
8 CosetsofSubgroupsandLagrange’sTheorem ................... 195
8.1 CosetsandTheirProperties ................................ 195
8.2 GeometricExamplesofCosets ............................. 198
8.3 Lagrange’sTheorem ...................................... 202
8.4 IndexofSubgroups ....................................... 207
8.5 ACountingPrincipleandDoubleCosets .................... 209
8.6 Worked-OutProblems .................................... 212
8.7 SupplementaryExercises .................................. 214
9 NormalSubgroupsandFactorGroups .......................... 217
9.1 NormalSubgroups ....................................... 217
9.2 FactorGroups ........................................... 222
9.3 Cauchy’sTheoremandClassEquation ...................... 225
9.4 Worked-OutProblems .................................... 229
9.5 SupplementaryExercises .................................. 231
Contents ix
10 SomeSpecialSubgroups ....................................... 233
10.1 CommutatorsandDerivedSubgroups ....................... 233
10.2 DerivedSubgroupsofSomeSpecialGroups ................. 237
10.3 MaximalSubgroups ...................................... 240
10.4 Worked-OutProblems .................................... 243
10.5 SupplementaryExercises .................................. 245
11 GroupHomomorphisms ....................................... 247
11.1 HomomorphismsandTheirProperties ....................... 247
11.2 IsomorphismTheorems ................................... 254
11.3 Cayley’sTheorem ........................................ 262
11.4 Automorphisms .......................................... 265
11.5 CharacteristicSubgroups .................................. 268
11.6 AnotherViewofLinearGroups ............................ 270
11.7 Worked-OutProblems .................................... 273
11.8 SupplementaryExercises .................................. 279
References ........................................................ 283
Index ............................................................. 287
About the Author
BijanDavvazisProfessorattheDepartmentofMathematics,YazdUniversity,Iran.
Earlier, he served as the Head of the Department of Mathematics (1998–2002),
ChairmanoftheFacultyofScience(2004–2006),andVice-PresidentforResearch
(2006–2008) at Yazd University, Iran. He earned his Ph.D. in Mathematics with a
thesison“TopicsinAlgebraicHyperstructures”fromTarbiatModarresUniversity,
Iran,andcompletedhisM.Sc.inMathematicsfromtheUniversityofTehran,Iran.
Hisareasofinterestincludealgebra,algebraichyperstructures,roughsetsandfuzzy
logic.Ontheeditorialboardsfor25mathematicaljournals,Prof.Davvazhasauthored
6booksandover600researchpapers,especiallyonalgebra,fuzzylogic,algebraic
hyperstructuresandtheirapplications.
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