Schaums outline of theory and problems of abstract algebra

310 Pages·2003·2.66 MB·English

by  Lloyd Jaisingh

Theory and Problems of
ABSTRACT
ALGEBRA
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Theory and Problems of
ABSTRACT
ALGEBRA
Second Edition
FRANK AYRES, Jr., Ph.D.
LLOYD R. JAISINGH
Professor of Mathematics
Morehead State University
Schaum’s Outline Series
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DOI: 10.1036/0071430989
Thisbookonalgebraicsystemsisdesignedtobeusedeitherasasupplementtocurrenttextsor
as a stand-alone text for a course in modern abstract algebra at the junior and/or senior levels.
In addition, graduate students can use this book as a source for review. As such, this book is
intendedtoprovideasolidfoundationforfuturestudyofavarietyofsystemsratherthantobe
a study in depth of any one or more.
The basic ingredients of algebraic systems–sets of elements, relations, operations, and
mappings–are discussed in the first two chapters. The format established for this book is as
follows:
. a simple and concise presentation of each topic
. a wide variety of familiar examples
. proofs of most theorems included among the solved problems
. a carefully selected set of supplementary exercises
Inthis upgrade,the text has made an effort to use standard notations for the set of natural
numbers, the set of integers, the set of rational numbers, and the set of real numbers. In
addition, definitions are highlighted rather than being embedded in the prose of the text.
Also, a new chapter (Chapter 10) has been added to the text. It gives a very brief discussion
of Sylow Theorems and the Galois group.
The text starts with the Peano postulates for the natural numbers in Chapter 3, with the
various number systems of elementary algebra being constructed and their salient properties
discussed. This not only introduces the reader to a detailed and rigorous development of these
number systems but also provides the reader with much needed practice for the reasoning
behind the properties of the abstract systems which follow.
The first abstract algebraic system–the Group–is considered in Chapter 9. Cosets of a
subgroup, invariant subgroups, and their quotient groups are investigated as well. Chapter 9
ends with the Jordan–Ho¨lder Theorem for finite groups.
Rings, Integral Domains Division Rings, Fields are discussed in Chapters 11–12 while
Polynomials over rings and fields are then considered in Chapter 13. Throughout these
chapters, considerable attention is given to finite rings.
Vector spaces are introduced in Chapter 14. The algebra of linear transformations on a
vectorspaceoffinitedimensionleadsnaturallytothealgebraofmatrices(Chapter15).Matrices
are then used to solve systems of linear equations and, thus provide simpler solutions to
a number of problems connected to vector spaces. Matrix polynomials are discussed in
v
Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.
vi PREFACE
Chapter 16 as an example of a non-commutative polynomial ring. The characteristic
polynomial of a square matrix over a field is then defined. The characteristic roots
and associated invariant vectors of real symmetric matrices are used to reduce the equations
of conics and quadric surfaces to standard form. Linear algebras are formally defined in
Chapter 17 and other examples briefly considered.
In the final chapter (Chapter 18), Boolean algebras are introduced and important
applications to simple electric circuits are discussed.
Theco-authorwishestothankthestaffoftheSchaum’sOutlinesgroup,especiallyBarbara
Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author
wishes to thank the estate of Dr. Frank Ayres, Jr. for allowing me to help upgrade the
original text.
LLOYD R. JAISINGH
For more information about this title, click here
PART I SETS AND RELATIONS
Chapter 1 Sets 1
Introduction 1
1.1 Sets 1
1.2 Equal Sets 2
1.3 Subsets of a Set 2
1.4 Universal Sets 3
1.5 Intersection and Union of Sets 4
1.6 Venn Diagrams 4
1.7 Operations with Sets 5
1.8 The Product Set 6
1.9 Mappings 7
1.10 One-to-One Mappings 9
1.11 One-to-One Mapping of a Set onto Itself 10
Solved Problems 11
Supplementary Problems 15
Chapter 2 Relations and Operations 18
Introduction 18
2.1 Relations 18
2.2 Properties of Binary Relations 19
2.3 Equivalence Relations 19
2.4 Equivalence Sets 20
2.5 Ordering in Sets 21
2.6 Operations 22
2.7 Types of Binary Operations 23
2.8 Well-Defined Operations 25
2.9 Isomorphisms 25
vii
viii CONTENTS
2.10 Permutations 27
2.11 Transpositions 29
2.12 Algebraic Systems 30
Solved Problems 30
Supplementary Problems 34
PART II NUMBER SYSTEMS
Chapter 3 The Natural Numbers 37
Introduction 37
3.1 The Peano Postulates 37
3.2 Addition on N 37
3.3 Multiplication on N 38
3.4 Mathematical Induction 38
3.5 The Order Relations 39
3.6 Multiples and Powers 40
3.7 Isomorphic Sets 41
Solved Problems 41
Supplementary Problems 44
Chapter 4 The Integers 46
Introduction 46
4.1 Binary Relation (cid:1) 46
4.2 Addition and Multiplication on J 47
4.3 The Positive Integers 47
4.4 Zero and Negative Integers 48
4.5 The Integers 48
4.6 Order Relations 49
4.7 Subtraction ‘‘(cid:2)’’ 50
4.8 Absolute Value jaj 50
4.9 Addition and Multiplication on Z 51
4.10 Other Properties of Integers 51
Solved Problems 52
Supplementary Problems 56
Chapter 5 Some Properties of Integers 58
Introduction 58
5.1 Divisors 58
5.2 Primes 58
5.3 Greatest Common Divisor 59
5.4 Relatively Prime Integers 61
5.5 Prime Factors 62
CONTENTS ix
5.6 Congruences 62
5.7 The Algebra of Residue Classes 63
5.8 Linear Congruences 64
5.9 Positional Notation for Integers 64
Solved Problems 65
Supplementary Problems 68
Chapter 6 The Rational Numbers 71
Introduction 71
6.1 The Rational Numbers 71
6.2 Addition and Multiplication 71
6.3 Subtraction and Division 72
6.4 Replacement 72
6.5 Order Relations 72
6.6 Reduction to Lowest Terms 73
6.7 Decimal Representation 73
Solved Problems 75
Supplementary Problems 76
Chapter 7 The Real Numbers 78
Introduction 78
7.1 Dedekind Cuts 79
7.2 Positive Cuts 80
7.3 Multiplicative Inverses 81
7.4 Additive Inverses 81
7.5 Multiplication on K 82
7.6 Subtraction and Division 82
7.7 Order Relations 83
7.8 Properties of the Real Numbers 83
Solved Problems 85
Supplementary Problems 87
Chapter 8 The Complex Numbers 89
Introduction 89
8.1 Addition and Multiplication on C 89
8.2 Properties of Complex Numbers 89
8.3 Subtraction and Division on C 90
8.4 Trigonometric Representation 91
8.5 Roots 92
8.6 Primitive Roots of Unity 93
Solved Problems 94
Supplementary Problems 95
x CONTENTS
PART III GROUPS, RINGS AND FIELDS
Chapter 9 Groups 98
Introduction 98
9.1 Groups 98
9.2 Simple Properties of Groups 99
9.3 Subgroups 100
9.4 Cyclic Groups 100
9.5 Permutation Groups 101
9.6 Homomorphisms 101
9.7 Isomorphisms 102
9.8 Cosets 103
9.9 Invariant Subgroups 105
9.10 Quotient Groups 106
9.11 Product of Subgroups 107
9.12 Composition Series 107
Solved Problems 109
Supplementary Problems 116
Chapter 10 Further Topics on Group Theory 122
Introduction 122
10.1 Cauchy’s Theorem for Groups 122
10.2 Groups of Order 2p and p2 122
10.3 The Sylow Theorems 123
10.4 Galois Group 124
Solved Problems 125
Supplementary Problems 126
Chapter 11 Rings 128
Introduction 128
11.1 Rings 128
11.2 Properties of Rings 129
11.3 Subrings 130
11.4 Types of Rings 130
11.5 Characteristic 130
11.6 Divisors of Zero 131
11.7 Homomorphisms and Isomorphisms 131
11.8 Ideals 132
11.9 Principal Ideals 133
11.10 Prime and Maximal Ideals 134
11.11 Quotient Rings 134
11.12 Euclidean Rings 135
Solved Problems 136
Supplementary Problems 139