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Elementary Abstract Algebra PDF

105 Pages·2007·0.52 MB·English
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Elementary Abstract Algebra W(cid:2) Edwin Clark Department of Mathematics University of South Florida (cid:2)Last revised(cid:3) December (cid:4)(cid:5)(cid:6) (cid:4)(cid:7)(cid:7)(cid:8)(cid:9) Copyright c (cid:2)(cid:3)(cid:3)(cid:4) by W(cid:5) Edwin Clark (cid:0) All rights reserved(cid:5) i ii Preface This book is intended for a one semester introduction to abstract algebra(cid:5) Most introductory textbooks on abstract algebra are written with a two semester course in mind(cid:5) See(cid:6) for example(cid:6) the books listed in the Bibli(cid:7) ography below(cid:5) These books are listed in approximate order of increasing di(cid:8)culty(cid:5) A search of the library using the keywords abstract algebra or modern algebra will produce a much longer list of such books(cid:5) Some will be readable by the beginner(cid:6) some willbe quite advanced and willbe di(cid:8)cult to understand without extensive background(cid:5) A search on the keywords group and ring willalso produce a number of more specializedbooks on the subject matterofthiscourse(cid:5) If you wishto see what is goingon atthefrontierofthe subject(cid:6) you might take a look at some recent issues of the journals Journal of Algebra or Communications in Algebra which you will (cid:9)nd in our library(cid:5) Instead of spending a lot of time going over background material(cid:6) we go directly into the primary subject matter(cid:5) We discuss proof methods and necessary background as the need arises(cid:5) Nevertheless(cid:6) you should at least skim the appendices where some of this material can be found so that you will know where to look if you need some fact or technique(cid:5) Since we only have one semester(cid:6) we do not have time to discuss any of the many applications of abstract algebra(cid:5) Students who are curious about applications will (cid:9)nd some mentioned in Fraleigh (cid:10)(cid:2)(cid:11) and Gallian (cid:10)(cid:12)(cid:11)(cid:5) Many more applications are discussed in Birkho(cid:13) and Bartee (cid:10)(cid:14)(cid:11) and in Dornho(cid:13) and Horn (cid:10)(cid:15)(cid:11)(cid:5) Although abstract algebra has many applications in engineering(cid:6) com(cid:7) puter science and physics(cid:6) the thought processes one learns in this course maybemorevaluablethan speci(cid:9)csubject matter(cid:5) In thiscourse(cid:6) onelearns(cid:6) perhaps for the (cid:9)rst time(cid:6) how mathematics is organized in a rigorous man(cid:7) ner(cid:5) This approach(cid:6) the axiomatic method(cid:6) emphasizes examples(cid:6) de(cid:9)nitions(cid:6) theorems and proofs(cid:5) A great deal of importance is placed on understanding(cid:5) iii iv PREFACE Every detail should be understood(cid:5) Students should not expect to obtain this understanding without considerable e(cid:13)ort(cid:5) My advice is to learn each de(cid:9)nition as soon as it is covered in class (cid:16)if not earlier(cid:17) and to make a real e(cid:13)ort to solve each problem in the book before the solution is presented in class(cid:5) Many problems require the construction of a proof(cid:5) Even if you are not able to (cid:9)nd a particular proof(cid:6) the e(cid:13)ort spent trying to do so will help to increase your understanding of the proof when you see it(cid:5) With su(cid:8)cient e(cid:13)ort(cid:6) your abilitytosuccessfully prove statements on your own willincrease(cid:5) We assume that students have some familiarity with basic set theory(cid:6) linear algebra and calculus(cid:5) But very little of this nature will be needed(cid:5) To a great extent(cid:6) the course is self(cid:7)contained(cid:6) except for the requirement of a certain amount of mathematical maturity(cid:5) And(cid:6) hopefully(cid:6) the student(cid:18)s level of mathematical maturity will increase as the course progresses(cid:5) I will often use the symbol to indicate the end of a proof(cid:5) Or(cid:6) in some cases(cid:6) will indicate the fact that no more proof will be given(cid:5) In such cases the proof will either be assigned in the problems or a reference will be provided where the proof may be located(cid:5) This symbol was (cid:9)rst used for this purpose by the mathematician Paul Halmos(cid:5) Note(cid:19) when teaching this course I usually present in class lots of hints and(cid:20)or outlines of solutions for the less routine problems(cid:5) This version includes a number of improvements and additions suggested by my colleague Mil(cid:21)e Kraj(cid:22)cevski(cid:5) Bibliography (cid:10)(cid:2)(cid:11) J(cid:5) B(cid:5) Fraleigh(cid:6) A First Course in Abstract Algebra(cid:6) (cid:16)Fifth Edition(cid:17)(cid:6) Addison(cid:7)Wesley(cid:6) (cid:2)(cid:3)(cid:3)(cid:23)(cid:5) (cid:10)(cid:12)(cid:11) J(cid:5) A(cid:5) Gallian(cid:6) Contemporary Abstract Algebra(cid:6) (cid:16)Third Edition(cid:17)(cid:6) D(cid:5)C(cid:5) Heath(cid:6) (cid:2)(cid:3)(cid:3)(cid:23)(cid:5) (cid:10)(cid:24)(cid:11) G(cid:5) Birkho(cid:13) and S(cid:5) MacLane(cid:6) A Survey of Modern Algebra(cid:6) A(cid:5) K(cid:5) Peters Ltd(cid:5)(cid:6) (cid:2)(cid:3)(cid:3)(cid:25)(cid:5) (cid:10)(cid:23)(cid:11) I(cid:5) N(cid:5) Herstein(cid:6) Topics in Algebra(cid:6) (cid:16)Second Edition(cid:17)(cid:6) Blaisdell(cid:6) (cid:2)(cid:3)(cid:25)(cid:14)(cid:5) (cid:10)(cid:14)(cid:11) G(cid:5) D(cid:5) Birkho(cid:13) and T(cid:5) C(cid:5) Bartee(cid:6) Modern Applied Algebra(cid:6) McGraw(cid:7)Hill Book Company(cid:6) (cid:2)(cid:3)(cid:25)(cid:26)(cid:5) (cid:10)(cid:15)(cid:11) L(cid:5) Dornho(cid:13) and F(cid:5) Hohn(cid:6) Applied Modern Algebra(cid:6) Macmillan(cid:6) (cid:2)(cid:3)(cid:25)(cid:4)(cid:5) (cid:10)(cid:25)(cid:11) B(cid:5) L(cid:5) Van der Waerden(cid:6) Modern Algebra(cid:6) (cid:16)Seventh Edition(cid:6) (cid:12) vols(cid:17)(cid:6) Fredrick Ungar Publishing Co(cid:5)(cid:6) (cid:2)(cid:3)(cid:25)(cid:26)(cid:5) (cid:10)(cid:4)(cid:11) T(cid:5) W(cid:5) Hungerford(cid:6) Algebra(cid:6) Springer Verlag(cid:6) (cid:2)(cid:3)(cid:4)(cid:26)(cid:5) (cid:10)(cid:3)(cid:11) N(cid:5) Jacobson(cid:6) Basic Algebra I and II(cid:6) (cid:16)Second Edition(cid:6) (cid:12) vols(cid:17)(cid:6) W(cid:5) H(cid:5) Freeman and Company(cid:6) (cid:2)(cid:3)(cid:4)(cid:3)(cid:5) (cid:10)(cid:2)(cid:26)(cid:11) S(cid:5) Lang(cid:6) Algebra(cid:6) Addison(cid:7)Wesley(cid:6) (cid:16)Third Edition(cid:17)(cid:6) (cid:2)(cid:3)(cid:3)(cid:12)(cid:5) v vi BIBLIOGRAPHY Contents Preface iii (cid:2) Binary Operations (cid:2) (cid:3) Introduction to Groups (cid:4) (cid:5) The Symmetric Groups (cid:2)(cid:6) (cid:7) Subgroups (cid:5)(cid:2) (cid:8) The Group of Units of Zn (cid:5)(cid:6) (cid:9) Direct Products of Groups (cid:5)(cid:4) (cid:6) Isomorphism of Groups (cid:7)(cid:2) (cid:10) Cosets and Lagrange(cid:11)s Theorem (cid:7)(cid:4) (cid:4) Introduction to Ring Theory (cid:8)(cid:8) (cid:2)(cid:12) Axiomatic Treatment of R(cid:13) N(cid:13) Z(cid:13) Q and C (cid:9)(cid:2) (cid:2)(cid:2) The Quaternions (cid:6)(cid:2) (cid:2)(cid:3) The Circle Group (cid:6)(cid:8) A Some Rules of Logic (cid:10)(cid:2) B Functions (cid:10)(cid:8) vii viii CONTENTS C Elementary Number Theory (cid:10)(cid:4) D Partitions and Equivalence Relations (cid:4)(cid:5) Chapter (cid:2) Binary Operations The most basic de(cid:9)nition in this course is the following(cid:19) De(cid:14)nition (cid:2)(cid:15)(cid:2) A binary operation on a set S is a function from S S (cid:2) (cid:3) to S(cid:2) If (cid:16)a(cid:2)b(cid:17) S S then we write a b to indicate the image of the element (cid:4) (cid:3) (cid:2) (cid:16)a(cid:2)b(cid:17) under the function (cid:2) (cid:2) The following lemma explains in more detail exactly what this de(cid:9)nition means(cid:5) Lemma (cid:2)(cid:15)(cid:2) A binary operation on a set S is a rule for combining two (cid:2) elements of S to produce a third element of S(cid:2) This rule must satisfy the following conditions(cid:3) (cid:16)a(cid:17) a S and b S (cid:27) a b S(cid:2) (cid:10)S is closed under (cid:5)(cid:11) (cid:4) (cid:4) (cid:5) (cid:2) (cid:4) (cid:2) (cid:16)b(cid:17) For all a(cid:2)b(cid:2)c(cid:2)d in S a (cid:27) c and b (cid:27) d (cid:27) a b (cid:27) c d(cid:3) (cid:10)Substitution is permissible(cid:5)(cid:11) (cid:5) (cid:2) (cid:2) (cid:16)c(cid:17) For all a(cid:2)b(cid:2)c(cid:2)d in S a (cid:27) b (cid:27) a c (cid:27) b c(cid:2) (cid:5) (cid:2) (cid:2) (cid:16)d(cid:17) For all a(cid:2)b(cid:2)c(cid:2)d in S c (cid:27) d (cid:27) a c (cid:27) a d(cid:2) (cid:5) (cid:2) (cid:2) Proof Recall that a function f from set A to set B is a rule which assigns to each element x A an element(cid:6) usually denoted by f(cid:16)x(cid:17)(cid:6) in the set B(cid:5) (cid:4) Moreover(cid:6) this rule must satisfy the condition x (cid:27) y (cid:27) f(cid:16)x(cid:17) (cid:27) f(cid:16)y(cid:17) (cid:16)(cid:2)(cid:5)(cid:2)(cid:17) (cid:5) (cid:2) (cid:12) CHAPTER (cid:2)(cid:3) BINARY OPERATIONS On the other hand(cid:6) the Cartesian product S S consists of the set of all (cid:3) ordered pairs (cid:16)a(cid:2)b(cid:17) where a(cid:2)b S(cid:5) Equality of ordered pairs is de(cid:9)ned by (cid:4) the rule a (cid:27) c and b (cid:27) d (cid:16)a(cid:2)b(cid:17) (cid:27) (cid:16)c(cid:2)d(cid:17)(cid:3) (cid:16)(cid:2)(cid:5)(cid:12)(cid:17) (cid:6)(cid:5) Now in this case we assume that is a function from the set S S to the (cid:2) (cid:3) set S and instead of writing (cid:16)a(cid:2)b(cid:17) we write a b(cid:5) Now(cid:6) if a(cid:2)b S then (cid:2) (cid:2) (cid:4) (cid:16)a(cid:2)b(cid:17) S S(cid:5) So the rule assigns to (cid:16)a(cid:2)b(cid:17) the element a b S(cid:5) This (cid:4) (cid:3) (cid:2) (cid:2) (cid:4) establishes (cid:16)a(cid:17)(cid:5) Now implication (cid:16)(cid:2)(cid:5)(cid:2)(cid:17) becomes (cid:16)a(cid:2)b(cid:17) (cid:27) (cid:16)c(cid:2)d(cid:17) (cid:27) a b (cid:27) c d(cid:3) (cid:16)(cid:2)(cid:5)(cid:24)(cid:17) (cid:5) (cid:2) (cid:2) From (cid:16)(cid:2)(cid:5)(cid:12)(cid:17) and (cid:16)(cid:2)(cid:5)(cid:24)(cid:17) we obtain a (cid:27) c and b (cid:27) d (cid:27) a b (cid:27) c d(cid:3) (cid:16)(cid:2)(cid:5)(cid:23)(cid:17) (cid:5) (cid:2) (cid:2) This establishes (cid:16)b(cid:17)(cid:5) To prove (cid:16)c(cid:17) we assume that a (cid:27) b(cid:5) By re(cid:28)exivity of equality(cid:6) we have for all c S that c (cid:27) c(cid:5) Thus we have a (cid:27) b and c (cid:27) c and it follows from (cid:4) part (cid:16)b(cid:17) that a c (cid:27) b c(cid:6) as desired(cid:5) The proof of (cid:16)d(cid:17) is similar(cid:5) (cid:2) (cid:2) Remarks In part (cid:16)a(cid:17) the order of a and b is important(cid:5) We do not assume that a b is the same as b a(cid:5) Although sometimes it may be true (cid:2) (cid:2) that a b (cid:27) b a(cid:6) it is not part of the de(cid:9)nition of binary operation(cid:5) (cid:2) (cid:2) Statement (cid:16)b(cid:17) says that if a (cid:27) c and b (cid:27) d(cid:6) we can substitute c for a and d for b in the expression a b and we obtainthe expression c d which is equal (cid:2) (cid:2) to a b(cid:5) One might not think that such a natural statement is necessary(cid:5) To (cid:2) see the need for it(cid:6) see Problem (cid:2)(cid:5)(cid:25) below(cid:5) Part (cid:16)c(cid:17) of the above lemma says that we can multiply both sides of an equation on the right by the the same element(cid:2) Part (cid:16)d(cid:17)(cid:6) says that we can multiply both sides of an equation on the left by the same element(cid:5) Binary operations are usually denoted by symbols such as (cid:29)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:4)(cid:2) (cid:2) (cid:2)(cid:0)(cid:2)(cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:7) (cid:2) (cid:3) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:7)(cid:7)(cid:7) Just as one often uses f for a generic function(cid:6) we use to indicate a generic (cid:2) binary operation(cid:5) Moreover(cid:6) if (cid:19) S S S is a given binary operation on (cid:2) (cid:3) (cid:18)

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This book is intended for a one semester introduction to abstract algebra. Most introductory textbooks on abstract algebra are written with a two semester course in mind. See, for example, the books listed in the Bibli- ography below. These books are listed in approximate order of increasing diffic
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