a S6 m6 U 5 F ‘ RBI Course inAlgebra Sets and groups Sets and groups A first course in algebra J. A. Green ProfessorofMathematics UniversityofWarwick Routledge&KeganPaul LondonandNewYork Firstpublishedin1965 Second,revisededitionI988by Routledge&KeganPaulLtd 11NewFetterLane,LondonEC4P4EE PublishedintheUSAby Routledge&KeganPaulInc. inassociationwithMethuenInc. 29West35thStreet,NewYork,NY10001 Setin10/12ptTimes byCotswoldTypesettingLtd,Gloucester andprintedinGreatBritain byRichardClayLtd, Bungay,Suffolk ©J.A.Green1965,1988 Nopartofthisbookmaybereproducedin anyformwithoutpermissionfromthepublisher exceptforthequotationofbriefpassages incriticism LibraryofCongressCataloginginPublicatData Green,J.A.(JamesAlexander) Setsandgroups. Includesindex. 1.Groups,Theoryof. 2.Settheory. I. Title. QA171.G68 1987 512 86—33887 BritishLibraryCIPDataalsoavailable ISBN0-7102-0557—0(c) 0-7102-1227-5(p) Contents Prefacetofirstedition Prefacetorevisededition 1 Sets 1.1 Sets 1.2 Subsets 1.3 Intersection 1.4 Union 1.5 Thealgebraofsets 1.6 Differenceandcomplement 1.7 Pairs.Productofsets 1.8 Setsofsets Exercises 2 Equivalencerelations 21 Relationsonaset 2.2 Equivalencerelations 2.3 Partitions 2.4 Equivalenceclasses 2.5 Congruenceofintegers 2.6 Algebraofcongruences Exercises 3 Maps 3.1 Maps 3.2 Equalityofmaps 3.3 Injective,surjective,bijectivemaps. Inversemaps 3.4 Productofmaps V CONTENTS 3.5 Identityrnaps 32 3.6 Productsofbijectivemaps 3.7 Permutations 34 3.8 Similarsets 37 Exercises 39 42 Groups 42 4.1 Binaryoperationsonaset 4.2 Commutativeandassociativeoperations 44 4.3 Unitsandzeros 47 4.4 Gruppoids,semigroupsandgroups 49 4.5 Examplesofgroups 51 4.6 Elementarytheoremsongroups 53 Exercises 55 Subgroups 57 5.1 Subsetsclosedtoanoperation 57 5.2 Subgroups 58 5.3 Subgroupgeneratedbyasubset 59 5.4 Cyclicgroups 61 5.5 Groupsactingonsets 62 5.6 Stabilizers 64 Exercises Cosets 68 6.1 Thequotientsetsofasubgroup 68 6.2 Mapsofquotientsets 70 6.3 Index.Transversals 71 6.4 Lagrange’stheorem 72 6.5 Orbitsandstabilizers 73 6.6 Conjugacyclasses.Centreofagroup 77 6.7 Normalsubgroups 81 6.8 Quotientgroups 82 Exercises 84 Homomorphisms 87 7.1 Homomorphisms 87 7.2 Somelemmasonhomomorphisms 88 con-rams vii 7.3 Isomorphism 90 7.4 Kernelandimage 91 7.5 Latticediagrams 94 7.6 Homomorphismsandsubgroups 96 7.7 Thesecondisomorphismtheorem 100 7.8 Directproductsanddirectsumsofgroups 102 Exercises 108 8 Ringsandfields 110 8.1 Definitionofaring.Examples 110 8.2 Elementarytheoremsofrings.Subrings 114 8.3 Integraldomains 118 8.4 Fields.Divisionrings 119 8.5 Polynomials 124 8.6 Homomorphisms.Isomorphismofrings 129 8.7 Ideals 132 8.8 Quotientrings 134 8.9 TheHomomorphismTheoremforrings 138 8.10 Principalidealsinacommutativering 139 8.11 TheDivisionTheoremforpolynomials 141 8.12 Polynomialsoverafield 145 8.13 DivisibilityinZandinF[X] 148 8.14 Euclid’salgorithm 155 Exercises 161 9 Vectorspacesandmatrices 165 9.1 Vectorspacesoverafield 165 9.2 Examplesofvectorspaces 168 9.3 Twogeometricinterpretationsofvectors 169 9.4 Subspaces 174 9.5 Linearcombinations.Spanningsets 176 9.6 Lineardependence.Basisofavectorspace 180 9.7 TheBasisTheorem.Dimension 184 9.8 Linearmaps.Isomorphismofvectorspaces 188 9.9 Matrices 192 9.10 Lawsofmatrixalgebra.TheringMAI") 197 9.11 Rowspaceofamatrix.Echelonmatrices 204 9.12 Systemsoflinearequations 209 viii CONTENIS 9.13 Matricesandlinearmaps 216 9.14 Invertiblematrices.ThegroupGL,(F) 226 Exercises 237 Tables 241 Listofnotations 247 Answerstoexercises 249 Index 255