MA136
             Introduction to Abstract Algebra
                       SamirSiksek
                       MathematicsInstitute
                       UniversityofWarwick
                                                                      Contents
                               ChapterI. Prologue                                                                         1
                                  I.1.   WhoAmI?                                                                          1
                                  I.2.   AJollyGoodRead!                                                                  1
                                  I.3.   Proofs                                                                           2
                                  I.4.   AcknowledgementsandCorrections                                                   2
                               ChapterII. AlgebraicReorientation                                                          3
                                  II.1.   Sets                                                                            3
                                  II.2.   BinaryOperations                                                                4
                                  II.3.   VectorOperations                                                                5
                                  II.4.   OperationsonPolynomials                                                         5
                                  II.5.   CompositionofFunctions                                                          6
                                  II.6.   CompositionTables                                                               7
                                  II.7.   CommutativityandAssociativity                                                   7
                                  II.8.   WherearetheProofs?                                                              9
                                  II.9.   TheQuaternionicNumberSystem(donotread)                                         10
                               ChapterIII.       Matrices—ReadOnYourOwn                                                  13
                                  III.1.   WhatareMatrices?                                                              13
                                  III.2.   MatrixOperations                                                              14
                                  III.3.   Wheredomatricescomefrom?                                                      16
                                  III.4.   Howtothinkaboutmatrices?                                                      17
                                  III.5.   WhyColumnVectors?                                                             19
                                  III.6.   Multiplicative Identity and Multiplicative Inverse                            20
                                  III.7.   Rotations                                                                     26
                               ChapterIV. Groups                                                                         27
                                  IV.1.   TheDefinitionofaGroup                                                           27
                                  IV.2.   First Examples(andNon-Examples)                                                27
                                  IV.3.   AbelianGroups                                                                  29
                                  IV.4.   SymmetriesofaSquare                                                            30
                               ChapterV. FirstTheorems                                                                   35
                                  V.1.   GettingRelaxedaboutNotation                                                     36
                                  V.2.   AdditiveNotation                                                                38
                               ChapterVI. MoreExamplesofGroups                                                           39
                                  VI.1.    MatrixGroupsI                                                                 39
                                  VI.2.    CongruenceClasses                                                             40
                                                                             i
                            ii                                 CONTENTS
                            ChapterVII. OrdersandLagrange’sTheorem                                         43
                              VII.1.   TheOrderofanElement                                                 43
                              VII.2.   Lagrange’sTheorem—Version1                                          46
                            ChapterVIII. Subgroups                                                         47
                              VIII.1.   WhatWereTheyAgain?                                                 47
                              VIII.2.   CriterionforaSubgroup                                              47
                              VIII.3.   RootsofUnity                                                       55
                              VIII.4.   MatrixGroupsII                                                     56
                              VIII.5.   Differential Equations                                             57
                              VIII.6.   Non-TrivialandProperSubgroups                                      58
                              VIII.7.   Lagrange’sTheorem—Version2                                         59
                            ChapterIX. CyclicGroupsandCyclicSubgroups                                      61
                              IX.1.   LagrangeRevisited                                                    64
                              IX.2.   SubgroupsofZ                                                         65
                            ChapterX. Isomorphisms                                                         67
                            ChapterXI. Cosets                                                              69
                              XI.1.   GeometricExamples                                                    70
                              XI.2.   SolvingEquations                                                     72
                              XI.3.   Index                                                                74
                              XI.4.   TheFirstInnermostSecretofCosets                                      74
                              XI.5.   TheSecondInnermostSecretofCosets                                     75
                              XI.6.   LagrangeSuper-Strength                                               76
                            ChapterXII. QuotientGroups                                                     79
                              XII.1.   CongruencesModuloSubgroups                                          79
                              XII.2.   CongruenceClassesandCosets                                          81
                              XII.3.   R/Z                                                                 82
                              XII.4.   R2/Z2                                                               83
                              XII.5.   R/Q                                                                 84
                              XII.6.   Well-DefinedandProofs                                                84
                            ChapterXIII. SymmetricGroups                                                   87
                              XIII.1.   Motivation                                                         87
                              XIII.2.   Injections, Surjections and Bijections                             88
                              XIII.3.   TheSymmetricGroup                                                  91
                              XIII.4.   Sn                                                                 91
                              XIII.5.   ANiceApplicationofLagrange’sTheorem                                94
                              XIII.6.   CycleNotation                                                      95
                              XIII.7.   PermutationsandTranspositions                                      99
                              XIII.8.   EvenandOddPermutations                                            100
                            ChapterXIV. Rings                                                             107
                              XIV.1.   Definition                                                          107
                              XIV.2.   Examples                                                           108
                                                            CONTENTS                                  iii
                             XIV.3.  Subrings                                                       110
                             XIV.4.  TheUnitGroupofaRing                                            112
                             XIV.5.  TheUnitGroupoftheGaussianIntegers                              115
                          ChapterXV. Fields                                                         119
                          ChapterXVI. CongruencesRevisited                                          121
                             XVI.1.  UnitsinZ/mZ                                                    121
                             XVI.2.  Fermat’sLittleTheorem                                          122
                             XVI.3.  Euler’s Theorem                                                123
                             XVI.4.   Vale Dicere                                                   124