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Introduction to Abstract Algebra PDF

207 Pages·1988·4.67 MB·English
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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.

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