Elementary Abstract Algebra - Mathematics - University of South

105 Pages·2007·0.52 MB·English

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)