Introduction to Abstract Algebra

207 Pages·1988·4.67 MB·English

by  Thomas A. Whitelaw B.Sc.

Introduction to
ABSTRACT ALGEBRA
Second Edition
THOMAS A. WHITELAW, B.Sc., Ph.D.
Lecturer in Mathematics
University of Glasgow
Blackie
Glasgow and London
Blackie and Son Limited,
Bishopbriggs, Glasgow G64 2NZ
7 Leicester Place, London WC2H 7BP
© 1988 Blackie and Son Ltd
First published 1978
This edition 1988
All rights reserved.
No part of this publication may be reproduced,
stored in a retrieval system, or transmitted,
in any form or by any means,
electronic, mechanical, recording or otherwise,
without prior permission of the Publishers.
British Library Cataloguing in Publication Data
Whitelaw, Thomas A.
Introduction to abstract algebra.--2nd ed.
1. Algebra, Abstract
I. Title
512'.02 QA162
ISBN 978-0-216-92259-4 ISBN 978-1-4615-7284-8 (eBook)
DOI10.1007/978-1-4615-7284-8
Photosetting by Advanced Filmsetters (Glasgow) Ltd.
Preface
THIS IS A TEXTBOOK FOR STUDENTS UNDERTAKING, OR ABOUT TO UNDERTAKE, A
first course in Abstract Algebra (e.g. second-year Honours students and
second-or third-year Ordinary Degree students in Scottish universities; first
year students in many English universities). The first edition of the book
proved helpful to many such students, and it is hoped that this second edition
may be found even more helpful. The main changes made in producing the
se co nd edition have been the inclusion of a whole new chapter on the
symmetrie group and the expansion of the existing sets of exercises, together
with updating and revising of the text.
The book is not an advanced treatise on group theory or on any kindred
part of mathematics. But, in contrast to many elementary textbooks, it does
penetrate far enough into Abstract Algebra to let the student see that the
subject has worthwhile insights to offer, and to introduce hirn to some of the
distinctive ways of thinking that produce interesting results. For example, in
group theory, the material covered includes cyclic groups, Lagrange's
theorem, homomorphisms, normal subgroups, quotient groups, and (in the
new chapter) the partition of the symmetrie group of degree n into conjugacy
classes and an intro duc ti on to the alternating group of degree n. Although
there is only one chapter on rings (and fields), it is a lengthy chapter and
covers a wide variety of ideas.
Experience shows that students find Abstract Algebra difficult to grasp if
they start with an inadequate understanding of basic topics (such as
equivalence relations, mappings, and the notations and vocabulary of set
theory.) Accordingly, as a preliminary to the chapters on groups, rings, etc.,
the first four chapters of this book deal with all the relevant details of those
basic topics, along with certain properties of the integers. This last item is
included to establish machinery (e.g. division algorithm) used in group theory
and elsewhere, and also to provide material which will serve to motivate and
illustrate some of the important concepts of Abstract Algebra. Chapter 1 also
includes some material that comes under the heading Logic, and consists of
explanations and advice about quantifiers, the use of the 'implies' sign, and
proof by contradiction. The book demands litth; in the way of pre-requisites,
though an elementary knowledge of complex numbers and matrices is as
sumed in some sections.
In writing this book, my aim throughout has been to make the exposition
of ideas as clear as possible, judged from the undergraduate student's point of
111
IV PREFACE
view. SO, for example, in considering mappings, relations, etc., I have avoided
presentations which, though satisfyingly neat and sophisticated to those who
have long understood the concepts, do little to communicate them to the
uninitiated.
The text includes worked examples and these have been chosen to reveal
important methods of argument and show how certain kinds of proof may be
set down. At the end of each chapter there is an am pie supply of graded
exercises, and partial solutions to these are at the back of the book. This
adjunct to the exercises will, if sensibly used, prove beneficial to the majority
of students.
I should like to record openly how much the merits ofthis book owe to the
tradition of care and enthusiasm in the teaching of mathematics that has long
existed in the Mathematics Department of the University of Glasgow. The
traditions I refer to nourished me as an undergraduate; and, more recently, in
working and conversing with my colleagues, I have learned much that has
enriched my ideas about how mathematics can and should be taught. Special
thanks go to my colleague Dr. J. B. Hickey for the help he has given during
the preparation of this book. His kindness and his attention to detail have far
exceeded all that I could have asked for.
T.A.W.
Contents
Chapter One SETS AND LOGIC
1. Some very general remarks 1
2. Introductory remarks on sets 2
3. Statements and conditions; quantifiers 3
4. The implies sign (=» 8
5. Proof by contradiction 11
6. Subsets 11
7. Unions and interseetions 15
8. Cartesian product of sets 18
EXERCISES 20
Chapter Two SOME PROPERTIES OF 7L
9. Introduction 22
10. The well-ordering principle 22
11. The division algorithm 24
12. Highest common factors and Euclid's algorithm 25
13. The fundamental theorem of arithmetic 29
14. Congruence modulo m (mE N) 30
EXERCISES 34
Chapter Three EQUIV ALENCE RELATIONS AND
EQUIV ALENCE CLASSES
15. Relations in general 35
16. Equivalence relations 36
17. Equivalence classes 37
18. Congruence classes 40
19. Properties of 7L as an algebraic system 42
m
EXERCISES 46
Chapter Four MAPPINGS
20. Introduction 47
v
vi CONTENTS
21. The image of a subset of the domain; surjections 49
22. Injections; bijections; inverse of a bijection 51
23. Restriction of a mapping 53
24. Composition of mappings 54
25. Some further results and examples on mappings 56
EXERCISES 59
Chapter Five SEMIGROUPS
26. Introduction 61
27. Binary operations 61
28. Associativity and commutativity 62
29. Semigroups: definition and examples 63
30. Powers of an element in a semigroup 64
31. Identity elements and inverses 65
32. Subsemigroups 66
EXERCISES 68
Chapter Six AN INTRODUCTION TO GROUPS
33. The definition of a group 69
34. Examples of groups 70
35. Elementary consequences of the group axioms 74
36. Subgroups 77
37. Some important general examples of subgroups 80
38. Period of an element 82
39. Cyc1ic groups 84
EXERCISES 87
Chapter Seven CO SETS AND LAGRANGE'S THEOREM ON
FINITE GROUPS
40. Introduction 89
41. Multiplication of subsets of a group 90
42. Another approach to cosets 91
43. Lagrange's theorem 96
44. Some consequences of Lagrange's theorem 98
102
EXERCISES
Chapter Eight HOMOMORPHISMS, NORMAL SUBGROUPS,
AND QUOTIENT GROUPS
45. Introduction 104
46. Isomorphie groups 104
CONTENTS VB
47. Homomorphisms and their elementary properties 107
48. Conjugacy 111
49. Normal subgroups 112
50. Quotient groups 115
51. The quotient group GjZ 117
52. The first isomorphism theorem 118
EXERCISES 121
Chapter Nine THE SYMMETRIC GROUP Sn 124
53. Introduction 124
54. Cycles 125
55. Products of disjoint cycles 127
56. Periods of elements of Sn 133
57. Conjugacy in Sn 134
58. Arrangements of the objects 1,2, ... , n 138
59. The alternating character, and alternating groups 140
60. The simplicity of As 144
EXERCISES 149
Chapter Ten RINGS 152
61. Introduction 152
62. The definition of a ring and its elementary consequences 152
63. Special types of ring and ring elements 154
64. Subrings and subfields 157
65. Ring homomorphisms 158
66. Ideals 159
67. Principal ideals in a commutative ring with a one 162
68. Factor rings 163
69. Characteristic of an integral domain or field 166
70. Factorization in an integral domain 168
71. Construction offields as factor rings 172
72. Polynomial rings over an integral domain 173
73. Some properties of F[X], where Fis a field 178
EXERCISES 180
BIBLIOGRAPHY 183
APPENDIX TO EXERCISES 184
INDEX 197
CHAPTER ONE
SETS AND LOGle
1. Some very general remarks
This opening seetion is intended as an introduction to the entire book and not
merely to the first chapter. It begins with an attempt to give brief answers to the
questions: What is Abstract Algebra? and Why is it a worthwhile subject of
study?
The thoughtful student may have realized that he has had to learn the
properties of several different algebraic systems-the system of integers, the
system of real numbers, the system of complex numbers, systems of matrices,
systems of vectors, etc. He would be correct to guess that the number of
algebraic systems he has to deal with will multiply rapidly as he proceeds
further in the study of mathematics. This being the case, it is desirable to try to
gain general insights about algebraic systems and to prove theorems that
apply to broad c1asses of algebraic systems, instead of devoting energy to the
study of a seemingly endless sequence of individual systems. If we can achieve
such general insights, we stand to gain a deeper and more unified understand
ing of numerous elementary matters such as properties of integers or
properties of matrices; and we shall be equipped to think intelligently about
new algebraic systems when we encounter them. Thus there are ample reasons
for believing it worthwhile (perhaps indeed essential) to attempt the study of
algebraic systems in general, as opposed to the separate study of many
individual algebraic systems. It is such study of algebraic systems in general
that is called Abstract Algebra, and this is the part of mathematics to which
this textbook provides an introduction.
The word abstract must not be taken to me an either difficult or unrelated to
the more tangible objects of mathematical study (such as numbers and
matrices). Instead, abstract indicates that the discussion will not be tied to one
particular concrete algebraic system (since generality of insight is the
paramount aim), while every general result obtained will have applications to
a variety of different concrete situations.
1
2 ABSTRACT ALGEBRA
In this book the study of abstract algebra proper begins at chapter five.
Before that, attention is given to preliminary material of fundamental
importance, induding sets, equivalence relations, and mappings.
This first chapter contains a survey of the main ideas used continually in the
discussion of sets. Interwoven with that are some remarks on elementary
logical matters such as quantifiers, the implies sign, and proof by con
tradiction. It is hoped that these remarks will help students both to digest
logical arguments that they read and to execute their own logical arguments.
Even so, a dear warning must be given that mastery of the art of logical
argument (as required in abstract algebra and comparable mathematical
disciplines) depends less on knowledge of any core offacts about logic than on
commitment to complete precision in the use of words and symbols.
2. Introductory remarks on sets
(a) Very often in mathematics we discuss sets or collections of objects and
think of each set as a single thing that may be denoted by a single letter (S or A
or ...) .
(b) Standard notations for familiar sets of numbers are:
N denoting the set of all natural numbers (i.e. positive integers);
7L denoting the set of all integers;
Q denoting the set of all rational numbers;
IR denoting the set of all real numbers;
C denoting the set of all complex numbers.
(c) Suppose that S is a set and x is an object. If xis one of the objects in the
set S, we say that x is a member of S (or x is an element of S, or x belongs to S)
and we indicate this by writing
XES.
In the contrary case, i.e. if x is not a member of S, we write
xrtS.
We also use the E symbol as an abbreviation for belonging to, as in the
phrase for every XE S (read for every x belonging to S).
(d) The set whose members are all the objects appearing in the list
Xl' X2' .•• , Xn and no others is denoted by
(e) A set S is called finite if only finitely many different objects are members
of S. Otherwise S is described as an infinite set.
If S is a finite set, the number of different objects in S is called the order (or
cardinality) of Sand denoted by ISI; e.g. if S = {1, 3, 5, 7}, then ISI = 4.