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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 FRANK AYRES, Jr., Ph.D., was formerly Professor and Head of the Department of Mathematics at Dickinson College, Carlisle, Pennsylvania. He is the author or coauthor of eight Schaum’s Outlines, including Calculus, Trigonometry, Differential Equations, and Modern Abstract Algebra. LLOYD R. JAISINGH is professor of Mathematics at Morehead State University (Kentucky) for the past eighteen years. He has taught mathematics and statistics during that time and has extensively integrated technology into the classroom. He has developed numerous activities that involve the MINITAB software, the EXCEL software, and the TI-83+ calculator. He was the recipient of the Outstanding Researcher and Teacher of the Year awards at Morehead State University. His most recent publication is the book entitled Statistics for the Utterly Confused, McGraw-Hill publishing. Copyright © 2004, 1965 by the McGraw-Hill Companies, Inc. All rights reserved. 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. ISBN: 978-0-07-143098-2 MHID: 0-07-143098-9 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-140327-6, MHID: 0-07-140327-2. All trademarks are trademarks of their respective owners. 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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 touse standard notations for the set ofnatural 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 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 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:2) 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:3)’’ 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