Sets & Groups: A First Course in Algebra

273 Pages·1988·10.422 MB·English

by  James Alexander Green (= Sandy Green)

a S6 m6 U 5
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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