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Theory and Problems of ABSTRACT ALGEBRA This page intentionally left blank. 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 McGRAW-HILL New York Chicago San Fransisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2004 1965 by McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-143098-9 The material in this eBook also appears in the print version of this title: 0-07-140327-2. All trademarks are trademarks of their respective owners. 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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