22m050_sg

The University of Iowa

Division of Continuing Education

Continuing Education Study Guide

for

MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I

College of Liberal Arts and Sciences

Mathematics

Course Prepared by

Daniel D. Anderson, Ph.D.

3 Semester Hours 20 Written Assignments

3 Examinations


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Continuing Education The University of Iowa

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MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I College of Liberal Arts and Sciences

Mathematics

Course Contents

Course Lessons

About the Coursewriter ............................................................................................ 5 Introduction ............................................................................................................. 6

About This Course ......................................................................................... 6 Required Course Materials ........................................................................... 6 Course Organization ..................................................................................... 7 Lesson Format ............................................................................................... 7 Web and E-mail ............................................................................................. 8 Examinations ................................................................................................ 9 Evaluation ................................................................................................... 10 How to Study ............................................................................................... 10

Unit 1 Preliminaries and the Definition of a Group.............................................. 13 Lesson 1 Set Theory ............................................................................................... 14

Written Assignment #1 ............................................................................... 15 Lesson 2 Mappings ................................................................................................ 15

Written Assignment #2 ............................................................................... 16 Lesson 3 A(S)—The Set of Bijections on S ............................................................. 17

Written Assignment #3 ................................................................................ 17 Lesson 4 The Integers ........................................................................................... 19

Written Assignment #4 ............................................................................... 21 Lesson 5 Mathematical Induction ........................................................................ 22

Written Assignment #5 ............................................................................... 22 Lesson 6 Complex Numbers ................................................................................. 24

Written Assignment #6 ............................................................................... 25 Lesson 7 Definitions and Examples of Groups ..................................................... 27

Written Assignment # 7 .............................................................................. 28 Lesson 8 Some Simple Properties of Groups ........................................................ 30

Written Assignment #8 ............................................................................... 31 Lesson 9 Self-Test #1 ............................................................................................. 32

Written Assignment #9 ............................................................................... 32 Examination #1 ........................................................................................... 32

Self-Test #1 ............................................................................................................. 34 Unit 2 Subgroups and Quotient Groups ............................................................... 35 Lesson 10 Subgroups ............................................................................................. 36

Written Assignment #10 ............................................................................. 37 Lesson 11 Lagrange's Theorem .............................................................................. 38


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Written Assignment #11 .............................................................................. 39 Lesson 12 Homomorphisms and Normal Subgroups ........................................... 40

Written Assignment #12 ............................................................................. 41 Lesson 13 Factor Groups ....................................................................................... 42

Written Assignment #13 ............................................................................. 43 Lesson 14 The Homomorphism Theorems ........................................................... 44

Written Assignment #14 ............................................................................. 45 Lesson 15 Self-Test #2 ........................................................................................... 46

Written Assignment #15 ............................................................................. 46 Examination #2 ........................................................................................... 46

Self-Test #2 ............................................................................................................ 48 Unit 3 The Symmetric Group and an Introduction to Rings ................................ 49 Lesson 16 Permutations and Cycles ...................................................................... 50

Written Assignment #16 ............................................................................. 51 Lesson 17 Odd and Even Permutations ................................................................ 53

Written Assignment #17 ............................................................................. 53 Lesson 18 Rings I ................................................................................................... 54

Written Assignment #18 ............................................................................. 55 Lesson 19 Rings II ................................................................................................. 56

Written Assignment #19 ............................................................................. 56 Lesson 20 Self-Test #3 .......................................................................................... 57

Written Assignment #20 ............................................................................ 57 Final Examination ....................................................................................... 57 Course Evaluation ....................................................................................... 58 Transcript .................................................................................................... 58

Self-Test #3 ............................................................................................................ 59

Continuing Education Policies and Instructions Be sure to read the Continuing Education (DCE) Policies and

Instructions before beginning this course. It is available on the ICON course site under Content; students who order the optional print material will receive a print copy by mail.


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About the Coursewriter

DANIEL D. ANDERSON, Professor of Mathematics, received his

Ph.D. in 1974 from the University of Chicago. He has taught at the Virginia

Polytechnic Institute and State University, the University of Missouri at

Columbia, and has been at The University of Iowa since 1977. Professor

Anderson has published over one hundred and fifty research articles in

commutative algebra, and has lectured on his research in Africa, Asia, and

Europe. His teaching experience ranges from courses in pre-algebra to

graduate courses in commutative ring theory. He is married, has a

daughter, son-in-law and grandson and enjoys collecting Iowa trade

tokens (tokens issued by pool halls, dairies, bars, general stories, etc.),

hiking, and picnicking.


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Introduction

About This Course

Introduction to Abstract Algebra I (22M:050), together with its

predecessor, Introduction to Linear Algebra (22M:027), constitutes the

"algebra" portion of an undergraduate mathematics major. This course is

an introduction to abstract or modern algebra with an emphasis on the

theory of groups. Rings are also briefly covered. The purpose of the course

is not just to introduce the student to groups and rings, but also to

introduce the student to the axiomatic and abstract point of view so

prevalent in modern mathematics, especially algebra, and to teach the

student to read and write proofs. This course should be of particular

interest to students who plan to become secondary teachers, to students

who plan to continue on in mathematics, and to students who would like a

taste of modern mathematics. Group theory is also a useful tool in many

other areas of study such as physics or engineering.

The course has as a prerequisite 22M:027 Introduction to Linear

Algebra or consent of the instructor. The course is self-contained and

adapted to independent study. Emphasis is placed on an ability to work

problems, both in the written assignments and in the examinations.

Required Course Materials

Materials Provided by the DCE

The following items may be accessed from the ICON course site

(under “Content”). They are also available in print from our office, and

may be purchased for an additional fee.

• Course Study Guide • Course Syllabus • Textbook and Materials Order Form • Policies and Instructions


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Textbooks/Course Materials to Purchase Independently

• Herstein, I. N. Abstract Algebra, third edition. New York: Wiley, 1999.

• Herstein, I. N. Student's Solution Manual to Abstract Algebra. New York: Macmillan Publishing Company, 1986. (selected chapters)

The course textbooks may be ordered from a local bookstore (see

Textbook and Materials Order Form) or from the vendor of your choice.

Note: If you purchase items from an alternate bookseller, it is imperative

that you obtain the correct editions.

Course Organization

The course consists of twenty lessons, divided into three units of

study. Unit I covers some preliminary material and ends with the

definition and some examples of groups. Unit II covers subgroups,

homomorphisms and normal subgroups, factor groups and the

Homomorphism Theorems. Unit III covers the symmetric group and a

very brief introduction to rings. The final lesson in each unit is a self-test,

designed to help prepare you for the exams.

Lesson Format

Each lesson consists of four parts. (1) A READING ASSIGNMENT in

the Herstein textbook. (2) A section of COMMENTS that elucidates the topic

of the reading assignment. You may wish to read the comments before you

complete the reading assignment to gain an overview of the material and

may also wish to review the comments after you complete the reading. (3)

A section of PRACTICE EXERCISES, consisting of selected problems drawn

from Herstein. At a minimum, you should work all of these problems

before completing the written assignment. The solutions to all the practice

problems (except those in Section 1.1) may be found in the Student's

Solution Manual. They should not be consulted until after you have

attempted the exercises. Do not submit the practice exercises to your


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instructor (unless, of course, you wish to pose a question about one of

them). (4) Finally, a WRITTEN ASSIGNMENT, consisting of selected

problems drawn from Herstein's Abstract Algebra. Work the problems

and submit all work, partial or complete. The assignments will be

graded by your instructor and returned to you. The Assignment

Identification Form must be submitted with each assignment. Note that

the written assignments for Lessons 9, 15, and 20 are self-tests (provided

in this study guide).

NOTE: You may turn in up to five assignments per week and may

turn in more provided you make special arrangements with the instructor.

Web and E-mail

This course is delivered on the World Wide Web via ICON (Iowa

Courses Online) http://icon.uiowa.edu/. You can access the course by

logging into ICON with your Hawk ID and password.

Online Tutorials

http://www.uiowa.edu/~online/tutorials/tutorial.html

View the online tutorials, which are provided in Flash format:

topics include instruction on using ICON, WebMail, Hawk ID Tools,

Security, and more. Please be aware that Continuing Education courses do

not use all of the components explained in the ICON tutorial.

Technical Support for Online Students

Technical assistance, including FAQs, software demos and

downloads, and contact information are provided on our technical support

pages: http://continuetolearn.uiowa.edu/ccp/sos/.

Hawk ID Help

http://hawkid.uiowa.edu/

Your Hawk ID and password are sent to you via email or mail the

first time you register at The University of Iowa. If you have forgotten

http://icon.uiowa.edu/

http://www.uiowa.edu/~online/tutorials/tutorial.html

http://continuetolearn.uiowa.edu/ccp/sos/

http://hawkid.uiowa.edu/


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your Hawk ID password or it has expired (after six months), you may call

the ITS Help Desk at the University and ask them to reset your password.

Please feel free to call our toll-free number (800.272.6430) and select the

phone routing option that connects you with the ITS Help Desk.

E-mail Alias

http://continuetolearn.uiowa.edu/ccp/sos/email.htm

A University of Iowa e-mail alias was created for you when you

enrolled in this course, if you didn't already have one. Your email alias

forwards messages to a specified email address, which can either be a UI

student email account or a non-UI account (e.g. Hotmail, Yahoo…etc.).

Once created, all subsequent e-mail contact from The University of Iowa

will go to your UI email alias. If you have not done so already, you should

login to your student account, i.e. ISIS http://isis.uiowa.edu/, then go to

My Uiowa/My Email and either request a UI email account or provide a

routing address.

E-mail is an official method of communication at The University of

Iowa; you are responsible for all information sent to your e-mail address of

record, and you may carry on official transactions with the University by

sending e-mail from your e-mail address of record. It is important that you

keep your email routing address in ISIS current if you prefer to use a non-

UI email account.

Examinations

There are three supervised examinations, a ninety-minute exam

following Unit I, a ninety-minute exam following Unit II (covering the

material in that unit), and a two-hour final following Unit III (covering all

course material, but with emphasis on material covered in Unit III). The

http://continuetolearn.uiowa.edu/ccp/sos/email.htm

http://isis.uiowa.edu/


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exams consist of problems to be solved. To arrange to take an exam, use

the Request for Examination form found at the end of each unit.

Prior to each examination, self-tests are provided in the study guide

(Lessons 9, 15, and 20). Complete the self-tests in the time indicated,

without using notes or other help—just as if they were supervised exams—

and submit them, along with the Assignment Identification Form, for

evaluation. The self-tests are intended to indicate to you and the instructor

areas where further study is advisable before you take the actual

examinations (which follow the same format as the self-tests).

Please read the information regarding exam scheduling and policies

posted on the ICON course Web site carefully. Students with access to the

Internet must use the ICON course Web site to submit exam requests

online. Students who do not have access to the internet may submit the

Examination Request Form located at the back of this Study Guide (print

version only).

Evaluation

You will be assigned a standard grade of A, B, C, D, or F in your

work in this course. Plus or minus will be assigned as appropriate.

There will be a possible total of 400 points for the course

distributed as follows:

Written Assignments First Examination Second Examination Final Examination

50 points 100 points 100 points 150 points

How to Study

The nature of independent study work necessarily places a greater

than usual study burden on the student, especially the mathematics


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student. Many of the concepts in abstract algebra are difficult to grasp

immediately, so don't be discouraged if the way is slow-going at first. You

can be sure you are not alone in this. The study guide has been designed to

help you as much as possible through the thorny patches in the textbook,

and, of course, you are encouraged to ask questions of your instructor if

you encounter material that is unclear to you. The more specific a question

is, the more easily it can be answered. Always cite the page number of the

textbook and cite the paragraph in question or the specific example

number or problem number.

Bear in mind that it is not realistic to expect to achieve full

comprehension of the material on a first reading. In studying each lesson,

read and reread the material. An initial skimming followed by a careful

reading, followed in turn by further study of difficult sentences or

paragraphs, should precede any attempt to work the problems. Even then,

further careful rereading will probably be necessary as the assigned

problems generate further questions in your mind.

Despite the investment of time required, try to work as many

problems as you can, since practice will improve your mathematical

facility. Mathematics cannot be learned passively.

The detailed solutions to all the practice exercises from each lesson

(with the exception of Section 1.1) are given in the Student's Solution

Manual, which also contains the solutions to several of the written

assignments and to many of the problems that were not assigned. Do not

look at the solutions until you have made an honest attempt at solving the

problem.

You will notice that the exercise sets usually consist of three types of

problems: easier problems, middle-level problems, and harder problems.

The bulk of the problems assigned will come from the easier problems

with some from the middle-level problems. This will also be representative

of the type or difficulty of problems on the tests. However, just because the


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problems are labeled "easier problems" does not mean that they are

necessarily easy. You should expect to have some difficulty with some of

the easier problems. On the other hand, you should not fail to look at an

exercise because it is labeled a "harder problem". Even if you fail to

completely solve it, you will learn something from your attempt. You are

encouraged to look at problems that were not assigned. The solutions to

almost all the middle-level and harder problems are given in the solution

manual.

Good luck with your studies!


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Unit 1 Preliminaries and the Definition of a Group

Lesson 1 Set Theory

Lesson 2 Mappings

Lesson 3 A(S)—The Set of Bijections on S

Lesson 4 The Integers

Lesson 5 Mathematical Induction

Lesson 6 Complex Numbers

Lesson 7 Definitions and Examples of Groups

Lesson 8 Some Simple Properties of Groups

Lesson 9 Self-Test #1

Examination #1


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Lesson 1 Set Theory

Reading Assignment and Comments

• Read the Preface of both the textbook and the solution manual. Also

Sections 1.1 and 1.2 of the textbook (Section n.m refers to the m th.

section of Chapter n of the textbook), pages 1–6.

Section 1.1 gives some remarks on what abstract algebra is.

Basically, modern or abstract algebra is the study of certain axiomatic

systems usually given by operations defined on elements of sets. You have

probably already encountered one such system: vector spaces. In this

course, you will study groups (see page 41 of the textbook for the

definition) and rings (see page 126 of the textbook for the definition).

One of the goals of this course is to learn to do proofs. It is

important to realize that when proving a statement you must "do the

general case". Just giving examples does not prove a theorem. In Exercise

2(a), page 3, you are asked to show that for ,baba −=∗ abba ∗≠∗

unless a = b. Here there are two things to prove: (1) if ,abba ∗=∗ then

,ba = (2) if ba = , then abba ∗=∗ . You must prove this for all ba and .

Giving particular examples does not constitute a proof. For example,

showing that taking 3and2 == ba gives 2332 ∗≠∗ is not a proof.

However, to prove a statement false, one particular example does suffice.

Consider the assertion: for each natural number ( ) 41, 2 +−= nnnfn is a

prime number. To see that this statement is false, note that

( ) ( ) 22 4111414141414141 =+−=+−=f and hence is not prime. It is

interesting to note that ( )nf is prime for 40,,2,1 =n .

Section 1.2 covers some basic aspects of set theory that will be used

throughout the course. While drawing Venn diagrams (see page 5) is a

useful way to visualize set operations, they do not constitute real proofs.

To show that two sets C and D are equal, you need to show that DC ⊂ and


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CD ⊂ . For example, to prove the set equality

( ) ( ) ( )CABACBA ∩∪∩=∪∩ in Exercise 9 (page 6) you must show that

( ) ( ) ( )CABACBA ∩∪∩⊂∪∩ and ( ) ( ) ( )CBACABA ∪∩⊂∩∪∩ .

Practice Exercises

• Page 2: 1, 3.

• Page 6: 3, 8, 10, 15, 17.

The solutions to almost all the practice exercises may be found in

the Student's Solution Manual.

Written Assignment #1

Instructions

Assignments can be submitted in print or via e-mail. See your

course syllabus (provided with print Study Guides and available on the

ICON course Web site) for detailed assignment submission instructions.

Description

• Page 3: 2.

• Page 6: 7, 9, 12, 14.

Lesson 2 Mappings


• Read Herstein: Section 1.3, pages 8–13.

The concept of function or mapping will be central to this course as

indeed it is to all of mathematics. While much of the material in this

section should be a review, it is nevertheless very important.

You are, of course, expected to know and understand all the

definitions given in the text. While you will not be asked to state


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definitions on the tests, I might, for example, ask you to show that the

composition of two bijections is a bijection.

Practice Exercises

• Page 13: 1, 5, 9, 12, 19, 28.


Instructions




Description

• Page 13: 2, 7, 14, 16, 17, 29.


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Lesson 3 A(S)—The Set of Bijections on S



This short section considers the set ( )SA of all bijections of S onto

itself. Lemma 1.4.1 says that ( )SA forms a group under composition (see

pages 40–41 of the textbook). The group ( )SA , where S is a finite set, will

be studied in greater detail in Chapter 3. As shown in the Example on page

18, all the familiar properties of multiplication of real numbers do not

carry over to ( )SA . For example, for ( )SAgf ∈, , we may have

( ) ,, 222 gffggffg ≠≠ and ggff ≠−1 .

In Exercise 10 (page 19) you are asked to show that if 3Sf ∈ , then

if =6 . Note that here f refers to a general function in 3S , not the

particular function f given at the bottom of page 17. Later, we will see that

if nSf ∈ , then if m = where !nm = .

In Exercise 17 (page 20) you are asked to show that .1 MMff =−

There is actually something to prove because, as noted in the previous

paragraph, gff 1− need not equal g. To show that ,1 MMff =− show that

MMff ⊆−1 (i.e., if ,Mg ∈ then Mgff ∈−1 ) and that .1MffM −⊆

Practice Exercises

• Page 19: 1, 2, 4, 9, 16.


Instructions





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Description

• Page 19: 3, 5, 10, 17, 23.


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Lesson 4 The Integers


• Read Herstein: Section 1.5 pages 21–28.

In this section, several properties or facts about the integers that

you have probably always taken for granted are stated and proved. The

first fact (which is an axiom), The Well-Ordering Principle, is that any

nonempty subset of natural numbers has a smallest element. Theorem

1.5.1 and its consequences are very important. Theorem 1.5.8 is often

called the Fundamental Theorem of Arithmetic.

Theorem 1.5.3 shows that if a and b are not both 0 , then their

greatest common divisor ( )bac ,= exists, is unique, and we can write c as a

linear combination of a and b, that is, there exist integers m and n with

nbmac += . While the proof given is an "existence proof", the example of

finding (24, 9) after the proof shows how to compute the greatest common

divisor ( )bac ,= using Theorem 1.5.1 and how to find an m and n with

nbmac += . We remark that many authors call Theorem 1.5.1 the "division

algorithm" and call the method of finding the greatest common divisor by

repeated applications of Theorem 1.5.1 the "Euclidean algorithm". (This

algorithm for finding the greatest common divisor actually appears in

Euclid's Elements which was written about 300 B.C.) By the way, you

should convince yourself ( )bac ,= , as defined in the textbook on page 23 is

actually the largest positive common divisor of a and b.

Let us look at this algorithm in more detail. Let a and b be integers

with b > 0. By Theorem 1.5.1, rbqa += where br <≤0 . If r = 0, then a =

bq and ( ) bba =, . Otherwise, again by Theorem 1.5.1, 11 rrqb += where

rr <≤ 10 . If 01 =r , then ( ) rba =, . Otherwise, continuing we get the

sequence


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rbqa +=

11 rrqb +=

221 rqrr +=

nnnn rqrr += −− 12

011 += +− nnn qrr

br <<0

brr <<< 10

brrr <<<< 120

brrr nn <<<<< −

10 .

Then ( )barn ,= . For 1−nn rr , so 21 −− =+ nnnnn rrqrr , and working

backwards we get that arbr nn and . But if ad and bd , then bqard −=

and working forwards we get finally that nrd . So nr does equal (a, b).

(Note that the sequence of remainders must eventually become 0 since

otherwise we would have an infinite decreasing sequence >>>> 21 rrrb of natural numbers, contradicting our assumption that

any set of natural numbers has a smallest element.) We can then write

( ) nbmarba n +==, by working backwards: =−= −− nnnn qrrr 12

( ) ( ) .1 2131232 nbmarqqrqqqrrr nnnnnnnnnn +==++−=−− −−−−−−−

The Division Algorithm and the Euclidean Algorithm also hold for

polynomials over a field F (such as the real numbers R ), see Theorem

4.5.5 and Theorem 4.5.7 (pages 156–157). Both the integers Z and

polynomials [ ]xF over a field F are examples of Euclidean rings, see page

162.

There are many other interesting properties of the integers and

especially prime numbers that are not discussed in this section; they are

studied in the area of mathematics called number theory. Let me mention

one such interesting result. In exercise 14, you are asked to show that there

are infinitely many primes of the form 4n + 3 and 6n + 5. These are very

special cases of a beautiful result due to Dirichlet. Let a and b be relatively

prime positive integers, then the sequence an + b; a + b, 2a + ab, 3a + b,


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…; contains infinitely many primes. The proof surprisingly requires the

use of complex numbers.

Practice Exercises

• Page 28: 2, 4, 7, 10, 17.


Instructions




Description

• Page 28: 1, 6, 8, 11, 13, 14.


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Lesson 5 Mathematical Induction


• Read Herstein: Section 1.6, 29–31.

This short section covers a very important proof technique—

mathematical induction. A proof by induction consists of two steps. First,

you must verify that the result is true for 1=n . This is usually easy. (But as

exercise 11 shows, it is a very important step.) Then using the truth of the

result for k, you must prove the result for 1+k .

Example 3 on page 30 is a pretty typical proof by induction: ( )nP is

the proposition =+++ n21 ( )121 +nn . It is easy to prove ( )1P . We then

assume ( ) kkP +++ 21: = ( )121 +kk . To prove ( )1+kP we add 1+k to

both sides to get 121 +++++ kk = ( ) 1121 +++ kkk . We must show

that ( ) =+++ 1121 kkk ( )( )21

2

1 ++ kk . But this is easy: ( ) ( ) =+++ 112

1 kkk

( ) ( ) ( ) ( )2112121

21

21 ++=+⋅++ kkkkk .

Exercise 10 gives another form of the Principal of Mathematical

Induction, sometimes called the Second Principal of Mathematical

Induction.

Practice Exercises

• Page 31: 1, 3, 9, 11.


Instructions





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Description

• Page 31: 2, 8, 10, 13.


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Lesson 6 Complex Numbers



Historically, the complex numbers were introduced so that all real

polynomials would have roots. For example, the quadratic equation x2 + 1

= 0 has no real solutions, but it has two complex solutions, namely i and –

i. The quadratic formula shows that any quadratic equation

02 =++ cbxax ( C∈cba ,, , 0≠a ) has two solutions: a

acbbr 2

41

2 −+−= and

a

acbbr 2

42

2 −−−= .

Alternatively, this says that ax2 + bx + c = a (x – r1 ) (x – r2).

(Remember that x – r is a factor of a polynomial if and only if r is a root!)

To solve (or factor) quadratic equations, we needed . = i 1− Do we need

to introduce more "imaginary" numbers to solve cubic equations, etc.? No!

The Fundamental Theorem of Algebra says that a polynomial p(x) =

a + + xa + xa n-nn 1

10 of degree n where 0,,,, 010 ≠∈ aaaa n C , has n

complex roots nrr ,,1 , or equivalently, that ( )xp breaks down into a

product of n linear factors: ( ) ( ) ( )nrxrxaxp −−= 10 .

Sometimes one writes . 1 + = + or 1 −− babia = i While this

notation is suggestive, one must exercise some care. The familiar rule

b a = ab for real numbers a, b ≥ 0 is not valid for negative numbers

since = )1()1( −− 11 = while . 1 = = = 1 1 2 −⋅−− iii

Perhaps more should be said about De Moivre's Theorem. If we

write =+= biaz )sincos( θθ i + r , then . ) sin + (cos r = z θθ nin nn For

example,


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( ) ( )( )( )

( ). i =

i =

i =

i = i

4424

sincos2

sincos21

22

22

45

45

44

55

25

−−

−−

+

+++

ππ

ππ

De Moivre's Theorem is also useful in computing roots of complex

numbers. If we write ,) i + ( r =z θθ sincos then z has n nth roots,

( ) ( )( ) . n , , = k , i + r nk +

nk + n 110sincos 22 −

πθπθ For example, the five

fifth roots of

( )44 sincos21 ππ ii +=+ are:

( ) , i + 202010 sincos2 ππ

( ) ( )( ) , + i + + 52

2052

2010 sincos2 ππππ

( ) ( )( ) , + i + + 54

2054


( ) ( )( ) andsincos2 56

2056

2010 , + i + + ππππ

( ) ( )( ) . + i + + 58

2058


Practice Exercises

• Page 37: 1, 5, 11, 20.


Instructions




Description

• Page 37: 2, 4, 13, 14, 22.


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Lesson 7 Definitions and Examples of Groups



Finally, we come to the definition of a group. Groups will occupy us

for the remainder of the course. You, of course, will be expected to know

and understand all the definitions in the text.

Let G be a nonempty set. A binary operation ∗ on G is a function

GGG →×∗ : . We denote the image ( )ba,∗ of ( )ba, by ba ∗ and usually

think of ∗ as a "product" of a and b. For example, if G = R, the set of real

numbers, then ×−+ and,, are all binary operations on R . From this point

of view, closure ( )GbaGba ∈∗⇒∈, is part of the definition of a binary

operation. But ÷ is not a binary operation on R since 0÷a is not defined.

A binary operation Gon∗ is associative if ( ) ( ) cbacba ∗∗=∗∗

for all Gcba ∈,, . Thus if ∗ is associative, we can just write cba ∗∗ . Note

that on =G R , ×+ and are associative while – is not. A set G with an

associative binary operation ∗ is called a semigroup. We usually just say

that ( )∗,G is a semigroup. If ( )∗,G is a semigroup with an identity element

( efffee ∗==∗ for all )Gf ∈ G is called a monoid. Finally, a monoid

( )∗,G is called a group if each element Gx∈ has an inverse (that is,

there is a Gy∈ with xyeyx ∗==∗ ).

Finite groups are often given by their multiplication tables. For

example, consider the set G ={e, a, b} with multiplication ∗ given by the

table:


A- 28

e a b e e a b a a b e b b e a

To find the product ba ∗ , we look in the row to the right of a and in

the column under b. Thus ba ∗ is the circled element e. In such a table for

a finite group each element occurs exactly once in each row and each

column. It would be instructive for you to construct such a table for the six

element group S3 defined on page 17 of the text.

You encountered a number of groups in linear algebra. The set of

mn × matrices over the reals forms an abelian group under matrix

addition while the set of invertible nn × matrices forms a group under

matrix multiplication. Under matrix addition, the identity element is the

nm × zero matrix while the identity element for matrix multiplication is

the nn × identity matrix In. Finally, if V is a real vector space, ( )+,V is an

abelian group.

Let me repeat that the concept of a group is fundamental for the

remainder of the course. Make sure that you have mastered this section

before proceeding.

Practice Exercises

• Page 46: 1, 2, 8, 13, 14, 18, 26, 28 (hard).

Written Assignment # 7

Instructions




Description

• Page 46: 3, 4, 15, 17, 25, 30.


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Lesson 8 Some Simple Properties of Groups



This very short section gives several important properties of groups.

It should also give you an idea of how one goes about proving results about

groups.

The problems in this section are not easy. In fact, most students will

find the majority of them difficult, even though their solutions (in the

solution manual) are short. The solution to each problem requires some

observation, insight, or "trick". However, hopefully these tricks will

become techniques that you will then have at your disposal. This is a

common phenomenon in mathematics. If an idea or trick occurs often

enough, it becomes a technique. (Didn't the method of substitution,

trigonometry substitution, or integration by parts at one time just seem

like a trick to you?) Be sure to give each problem an honest attempt before

you turn to the solution manual.

In this and future sections, we will drop the "∗ " from the product

ba ∗ of a and b and just write ab. However, do not write a/b or ba . In the

reals ba means 1−ab or ab 1− (they are of course equal), but in a nonabelian

group, we might well have abab 11 −− ≠ .

Practice Exercises

• Page 50: 1, 3.


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Instructions




Description

• Page 50: 2, 4, 6.


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• Review the material in Chapters 1 and Sections 1 and 2 of Chapter 2

of Herstein.

Do not begin the self-test on the following pages until your review is

completed. Regard the self-test as if it were a supervised examination and

take it within the given time limit and without any aids or references. After

you have completed the self-test, check over your answers and send your

answers to your instructor for grading.


Instructions




Description

• Complete the self-test which follows.

Examination #1

A supervised closed-book examination is scheduled following

Lesson 9. The examination consists of five problems and covers the

material in Unit I. You will be allowed ninety minutes to complete the

exam. The use of books, notes, calculator, or other aids is not permitted

during the examination. Scratch paper used during the exam must also be

submitted with the exam itself.






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version only).


A- 34

22M:050 Introduction to Abstract Algebra I

Self-Test #1

Work the following five problems. Each problem is worth 20 points.

Allow yourself ninety minutes for this self-test.

1. a. Let A, B, and C be sets. Prove that ( ) ( ) ( )CABACBA ∪∩∪=∩∪ .

b. If BAf →: and CBg →: are bijections, show that CAfg →: is

also a bijection.

2. Prove by mathematical induction that ( ) 1for11 −>+≥+ xnxx n .

3. a. Find the greatest common divisor ( ) 28and100of28,100=c and

find m and n so that 28100 ⋅+⋅= nmc .

b. Find the cube roots of 2 – 2i.

4. Show that the set G of 2 × 2 matrices of the form

ba 0

0 where a, b ∈

Q – {0} forms a group under the usual matrix product.

5. Show that a group G is abelian if and only if ( ) 222 baab = for all Gba ∈,

.

REMINDER: You must take Examination #1 before submitting subsequent written assignments, although you may work ahead on these assignments if you wish.


A- 35

Unit 2 Subgroups and Quotient Groups

Lesson 10 Subgroups

Lesson 11 Lagrange's Theorem

Lesson 12 Homomorphisms and Normal Subgroups

Lesson 13 Factor Groups

Lesson 14 The Homomorphism Theorems


Examination #2


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Lesson 10 Subgroups



In this lesson, you are introduced to the important notion of a

subgroup of a group. Recall from linear algebra that a subspace W of a

(real) vector space V is a nonempty subset W of V that is a vector space

with the same addition and scalar product as V. It was then shown that W

is a subspace of V if and only if W is closed under addition and scalar

product. In group theory, a subgroup of a group plays much the same role

as does a subspace of a vector space in linear algebra. Thus a nonempty

subset H of G is called a subgroup of G, if relative to the product in G, H

itself forms a group. As in analogy with a subspace being a nonempty

subset of a vector space closed under addition and scalar product, Lemma

2.3.1 shows that a nonempty subset H of a group G is a subgroup if H is

closed under the multiplication of ( )HabHbaG ∈⇒∈, and under

inverses ( )1−⇒∈ aHa . By Exercise 15 (page 55), this is equivalent to

HabHba ∈⇒∈ −1, . Note that if W is a subspace of a vector space V, then

( )+,W is a subgroup of ( )+,V .

In the next lesson, a fundamental result, Lagrange's Theorem, says

that if G is a finite group and H is a subgroup of G, then GH . Here G

denotes the number of elements of G. Don't miss the definition of a cyclic

group given between problems 11 and 12 on page 55. You should verify that

( )+,Z and the group B given in example 8 on page 53 are cyclic. As you

will see later, these are essentially the only cyclic groups.

Practice Exercises

• Page 54: 3, 4, 11, 13, 14, 18, 19.


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Instructions




Description

• Page 54: 1, 8, 10, 12, 15, 17, 20. (Hint: Try 3SG = ).


A- 38

Lesson 11 Lagrange's Theorem



The purpose of this section is to prove Lagrange's Theorem: If H is a

subgroup of a finite group G, then GH . This result has many

important consequences for groups (e.g., Theorems 2.4.3, 2.4.4 and 2.4.5)

as well as for number theory. Make sure that you carefully read the proof

of Lagrange's Theorem and the theorems in this section.

The converse of Lagrange's Theorem is false in general: if G is a

group of order n and m is a positive integer dividing n, G need not have a

subgroup of order m. The group A4 (defined on page 121) has order 12, but

has no subgroup of order 6. However, it is true that if G is an abelian

group of order n and m is a positive integer dividing n, then G does have a

subgroup of order m. Also, it follows from the Sylow Theorems (given in

section 2.11, but which we will not cover), that if G is a group of order n

and if p is a prime number with kp dividing n, then G has a subgroup of

order kp .

The notion of the order of an element is very important. An element

Ga∈ is said to have finite order if there is a positive integer n with

ea n = . (If no such positive integer exists, a is said to have infinite

order.) If a has finite order, the order of a is defined to be least positive

integer m with ea m = . Equivalently, m is the order of the cyclic subgroup

generated by a. Notice that the word order is used in two different ways.

If H is a subgroup of G and Gb∈ , then set { }HhhbHb ∈= is called

a right coset of H in G while the set { }HhbhbH ∈= is called a left

coset of H in G. Note that a coset is a subset of G. If the group G under

consideration is abelian and the group operation is denoted by + we often


A- 39

write { }HhhbHb ∈+=+ for the left coset bH. For example,

{ }ZnnZ ∈+=+ 2323 . Notice that if G is abelian, there is no difference

between right and left cosets. However, for G a nonabelian group and H a

subgroup of G, we may very well have GaHaaH ∈≠ for . (See Exercise 6,

page 64.)

It follows from Theorem 2.4.1 that if Ha and Hb are two right cosets

of a group G (H a subgroup of G), then either φ=∩= HbHaHbHa or .

Practice Exercises

• Page 63: 3, 8, 11, 12, 13, 28, 29.


Instructions




Description

• Page 63: 1, 4, 5, 6, 9, 22, 25, 31.


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Lesson 12 Homomorphisms and Normal Subgroups



In this lesson the important notions of homomorphism and normal

subgroup are introduced. Make sure that you thoroughly understand

them. Homomorphisms for groups play much the same role that linear

transformations do for vector spaces. For vector spaces, recall that a linear

transformation WVT →: where V and W are vector spaces is a map the

preserves the two vector space operations addition and scalar product,

that is ( ) ( ) ( )2121 ν+ν=ν+ν TTT and ( ) ( )11 να=αν TT . Thus, if

( ) ( )⋅′∗ ,and, GG are groups, a group homomorphism 1: GG →ϕ is

function that preserves the groups products ( ) ( ) ( )baba ϕϕϕ =∗ or if we

just denote product by juxtaposition, ( ) =abϕ ( ) ( )ba ϕϕ . Given a

homomorphism ( ) =′→ ϕϕ KGG ,: ( ){ }exGx =∈ ϕ , the kernel of ϕ , is a

normal subgroup of G. In the next lesson, it will be shown that every

normal subgroup is the kernel of some homomorphism. The image ( )Gϕ

of G is also a group. It inherits many properties from G. For example, if G

is abelian, so is ( )Gϕ .

By definition, N is a normal subgroup of G if NNaa ⊂−1 for each

Ga∈ . On page 71, it is shown that if GN , then we actually have

NNaa =−1 . So GN if and only if aNNa = for each Ga∈ , or

equivalently, if every left coset is a right coset. Normality can also be

viewed as a weakened form of commutativity. For Ga∈ and Nn∈ , we

cannot always conclude that anna = , but if, GN then nana ′= for some

Nn ∈′ since aNNa = .

Be sure to read the last paragraph before the problems on page 73.

There are many interesting exercises that I have not assigned that you


A- 41

might like to try. For example, Exercise 37 asks you to show that a

nonabelian group of order six must be isomorphic to S3, and Exercise 45

asks you to prove that any group of order p2, p a prime, must be abelian.

Cayley's Theorem states that every group is isomorphic to some

subgroup of ( )SA , for an appropriate set S. If G is finite with nG = , then

G is isomorphic to a subgroup of nS . The group nS will be studied in more

detail in Lessons 16 and 17.

Practice Exercises

• Page 73: 1, 3, 6, 8, 16, 24.


Instructions




Description

• Page 73: 2, (Hint: If 21: GGf → is an isomorphism, show that

121 : GGf →− is an isomorphism. If 21: GGf → and 32: GGg → are

isomorphisms, show that 2: GGfg → is an isomorphism.) 4, 7, 12,

17, 22, 28.


A- 42

Lesson 13 Factor Groups


• Read Herstein: Section 2.6, pages 77–82

Comments

In this lesson, you are introduced to the most important

construction in group theory: the factor group. The role model for the

factor group construction is Zn. In fact, Zn is nothing more than Z/N where

{ }Z∈= kknN . It might not be a bad idea to review the construction of Zn

given on pages 60 and 61 of Herstein.

Let GN . The factor group NG / is a set of sets! The elements of

NG / are the distinct right cosets of N in G. The coset { }NnnaNa ∈= can

also be viewed as the equivalence class [ ]a of a under the equivalence

relation a ~ b if and only if Hab ∈−1 . The product in NG / may either be

viewed as [ ] [ ] [ ]abba = or NabNaNb = . Make sure that you understand this

construction both from the point of view of equivalence classes (given on

pages 77–78) and from the point of view of coset (or set) products (given

on pages 79–80).

If GN , then Theorem 2.6.2 says that the map NGG /: →ψ given

by ( ) [ ]aa =ψ (or ( ) Naa =ψ ) is a homomorphism. This map is called the

natural homomorphism from G to NG / . Note that ψ is surjective and

( ) .NK =ϕ In the last lesson, it was shown that the kernel of a group

homomorphism was a normal subgroup. Conversely, any normal subgroup

N of G is the kernel of some homomorphism, namely the natural map

NGG /→ .

Suppose that G is a finite group. Note that NG / is the number of

left cosets of N in G. This number has already been given a name, the


A- 43

index of N and G, and was denoted by ( )NiG . The proof of Lagrange's

Theorem showed that ( ) NGNiG /= . Hence NGNG // = and hence is

a divisor of G .

Practice Exercises

• Page 82: 2, 4, 6, 7, 11, 13.


Instructions




Description

• Page 82: 1, 3, 8, 9, 12, 18.


A- 44

Lesson 14 The Homomorphism Theorems



The three homomorphism theorems (especially the first one) are

very powerful. For example, using the First Homomorphism Theorem we

can completely characterize all cyclic groups. Let G be a cyclic group with

generator a. Consider the map ( ) G→+ϕ ,: Z given by ( ) iai =ϕ . Then ϕ is

a surjective homomorphism. (Verify!) Let N be the kernel of ϕ . If a has

infinite order, { }0=N and ϕ is an isomorphism. So in the case G is

isomorphic to ( )+,Z . If ( ) ,∞<= nao then == ZnN { }Z∈znz . (Verify.)

Then by the First Homomorphism Theorem, G is isomorphic to ZZ n/ ,

that is, G is isomorphic to nZ . Exercises 2 and 3 are typical applications of

the first Homomorphism Theorem.

Let G be a group and let GK . What do the subgroups of KG /

look like? Of course, a subgroup of KG / being a subset of KG / is a

collection of right cosets. If H is a subgroup of G with GHK ⊂⊂ , then

HK , so we can form a factor group { }GhKhKH ∈=/ . Now KH / is a

subgroup of KF / (verify). Coversely, if H ′ is a subgroup of KG / by the

Correspondence Theorem (and the paragraph after its proof), KHH /=′

for some subgroup H of G with GHK ⊂⊂ . In fact, ( ){ }HaGaH ′∈π∈=

{ }HHaGa ′∈∈= where ( )KaaKGG →→π /: is the natural map.

Most students will probably view the homomorphism theorems as

rather abstract. This is true; but one of the purposes of this course is to

introduce you to modern abstract mathematics. You are expected to know

the proof of the First Homomorphism Theorem, but not of the other

Homomorphism Theorems.


A- 45

Practice Exercises

• Page 87: 1, 2, 3.


Instructions




Description

• Page 87: 4, 6, 7.


A- 46



• Review the material in Sections 2.3 through 2.7 of Herstein.




you have completed the self-test, check your answers and send your



Instructions




Description


Examination #2

A supervised closed-book examination is scheduled following

Lesson 15. The examination consists of five problems and covers the

material in Unit 2. You will be allowed ninety minutes to complete the

exam. The use of books, notes, calculator, or other aids is not permitted

during the examination. All scratch paper used during the exam will be

turned in with the exam itself.






A- 47


version only).


A- 48


Self-Test #2

Work the following five problems. Each problem is worth 20 points.

Allow yourself ninety minutes for this self-test.

1. Let H1 and H2 be subgroups of a group G.

a. Show that H1∩H2 is a subgroup of G.

b. Give an example to show that H1∪H2 need not be a subgroup.

2. State and prove the First Homomorphism Theorem.

3. Let Q be the rational numbers under addition. Show that every

element in the factor group Q/Z has finite order.

4. Show that if H is a subgroup of G with iG(H) = 2, then GH .

5. Show that a group of order p, p a prime, is cyclic.

REMINDER: You must take Examination #2 before submitting subsequent written assignments, although you may work ahead on these assignments if you wish.


A- 49

Unit 3 The Symmetric Group and an Introduction to Rings

Lesson 16 Permutations and Cycles

Lesson 17 Odd and Even Permutations

Lesson 18 Rings I

Lesson 19 Rings II


Final Examination


A- 50

Lesson 16 Permutations and Cycles


• Read Herstein: Sections 3.1 and 3.2, pages 108–117.

In this lesson, we go back and look at the group ( )SA of bijections

on a finite set S in more detail. When nS = , we denote ( )SA by nS and

call it the symmetric group of degree n. Elements of nS are called

permutations. Section 3.1 is introductory in nature.

In Section 3.2, the important notion of a k-cycle is introduced.

Theorem 3.2.2 states that every permutation in nS is the product of

disjoint cycles. Actually, while not stated, more is true. The representation

of a permutation as a product of disjoint cycles is unique up to the order of

factors. Theorem 3.2.5 gives the important result that every permutation is

a product of transpositions (but they aren't necessarily disjoint). While

this representation is not unique, we will see in the next lesson that the

number of transpositions is always either even or odd. Here is an intuitive

way to think about Theorem 3.2.5. Suppose you want to rearrange the

books on your bookshelf. You can do this by interchanging two books at a

time. The nonuniqueness follows since there are obviously many different

ways to carry out this rearrangement.

Let's list the elements of S3 and S4. We begin with S3, which has 6 = 3!

elements. Its elements are (1), (12), (13), (23), (123) = (13) (12), and (132) =

(12) (13).

S4 has 4! = 24 elements:

1 1-cycle: (1)

2346 = ⋅ 2-cycles: (12), (13), (14), (23), (24), (34)

(There are 4 choices for a and 3 choices for b in the 2-cycle (ab), but

we must divide by 2 since (ab) = (ba).)


A- 51

32348 = ⋅⋅ 3-cycles: (123), (124), (132), (134), (142), (143), (234), and

(243).

4

12346 ⋅⋅⋅= 4-cycles: (1234), (1243), (1324), (1342), (1423), and

(1432).

( ) = 2

122

34213 ⋅⋅ ⋅ products of disjoint 2-cycles: (12) (34), (13) (24),

(14) (23)

(The 21 in front is because disjoint 2-cycles commute; so, (12) (34) =

(34) (12), etc.)

This gives us 24 elements which uses up all of S4.

You are welcome to write out all 120 = 5! elements of S5. For

example, S5 has 1 1-cycle, 10225 = ⋅ 2-cycles, 203

345 = ⋅⋅ 3-cycles, 3042345 = ⋅⋅⋅

4-cycles and 24512345 = ⋅⋅⋅⋅ 5-cycles. There are ( ) ( ) 152

232

4521 = ⋅⋅ products of

disjoint 2-cycles such as (12) (34) and ( ) ( ) 202

12

3

345 = ⋅⋅⋅ products of disjoint

3- and 2-cycles such as (123) (45). This accounts for all 120 elements of S5.

Practice Exercises

• Page 110: 1, 4.

• Page 117: 2, 4, 10, 13, 18.


Instructions





A- 52

Description

• Page 110: 2, 3, 5.

• Page 117: 1, 3, 5, 8, 20.


A- 53

Lesson 17 Odd and Even Permutations



The main result of this section is Theorem 3.3.1, which states that a

permutation is either even or odd but not both. Since the product of two

even permutations is again even, the set nA of even permutations is a

subgroup of nA (here we have used the fact that a nonempty subset of a

finite group that is closed under products is a subgroup). However, not

only is nA a subgroup of nS , it is actually a normal subgroup of nS .

In Lesson 16, all the elements of S3, S4, and S5 were listed. You

should go back and determine A3, A4, and A5.

Practice Exercises

• Page 123: 2, 4, 5.


Instructions




Description

• Page 123: 1, 3, 6, 8. (Be sure to verify that the example given in the

Student's Solution Manual is actually a normal subgroup of 4A !)


A- 54

Lesson 18 Rings I



So far we have covered groups. In this lesson, you are introduced to

a second algebraic system, the ring. Briefly, a ring ( )⋅+,,R is a nonempty

set with two binary operations + (addition) and (multiplication) such that

(1) ( )+,R is an abelian group, (2) ( )⋅,R is a semigroup (that is, · is

associative), and (3) the distributive laws hold: ( ) cabacba ⋅+⋅=+⋅ and

( ) =⋅+ acb acab ⋅+⋅ . As with groups, we usually just denote ba ⋅ by ab

. This section contains a lot of definitions; you are expected to know them

all.

Familiar examples of rings include the integers Z , the rational

numbers Q , the integer mod nn Z and the set of nn × matrices over the

reals with the usual matrix addition and multiplication.

My area of research is commutative rings. A topic of special interest

to me is how results for integral domains carry over to commutative rings

with zero divisors.

You may have wondered where the name "ring" came from. Here is

one explanation. Some of the first rings considered come from algebraic

number theory. Here is an example. Let i+= 12γ and let [ ] =γ=ZR

{ }Z∈γ+ baba , . Note that R is indeed a ring. For ( ) =+=γ 22 1 i =−+ 121 i

( ) =−+= 2122 ii 2−γ . Hence, ( ) ( ) =++ γγ dcba ( ) =γ+γ++ 2bdbcadac

( ) ( )2−γ+γ++ bdbcadac ( ) ( ) .2 Rbdbcadbdac ∈γ+++−= So, since R is

closed under addition and multiplication it is a subring of C , the complex

numbers. The word "ring" supposed comes from the fact the 2γ cycles

("rings") back to the "lower degree term" .2−γ


A- 55

Practice Exercises

• Page 133: 1, 2, 4, 10, 13, 20.


Instructions




Description

• Page 133: 3, 5, 7, 16, 19.


A- 56

Lesson 19 Rings II


• Read Herstein: Section 4.2, pages 137–138

In this lesson, some simple results are derived from the ring

axioms.

Let R be a ring. In Lemma 4.24 it is shown that if xx =2 for all x in

R, then R is commutative, while Exercises 5 and 7 (page 139) show that if

xxxx == 43 or for all x in R, then R is commutative. Can you guess a more

general result? Here is one. If R is a ring and for each Rx∈ , there exists a

natural number ( ) 2≥xn ( ( )xn means that ( )xn depends on x ) with

( ) xx xn = , then R is commutative. This is an example of a commutativity

theorem. Commutativity theorems were a favorite topic of Professor

Herstein.

Practice Exercises

• Page 139: 4, 5, 7.


Instructions




Description

• Page 139: 1, 2, 3, 6.


A- 57



• Review the material in Chapters 1, 2 (Sections 1–7), 3, and 4

(Sections 1–2) of Herstein.




you have completed the self-test, check over your answers and send your



Instructions




Description


Final Examination

A supervised closed-book final examination is scheduled following

Lesson 20. The examination consists of eight problems and covers the

entire course, but with emphasis on the material in Unit 3. You will be

allowed two hours to complete the exam. The use of books, notes,

calculator, or other aids is not permitted during the examination. All

scratch paper used during the exam must be turned in with the exam itself.





A- 58



version only).

Course Evaluation

At the end of the semester you will receive an email inviting you to

submit a Course Evaluation. We would greatly appreciate it if you would

take a few moments to complete the Course Evaluation. Your evaluation

and additional written comments will help us improve the Continuing

Education courses we offer.

Students who complete their GIS course in two semesters will

receive the email invitation at the end of the second semester.

Transcript

http://registrar.uiowa.edu/transcripts/ Your final course grade will be entered on your permanent student

record at The University of Iowa. Official transcripts are available from the

Office of the Registrar, and may be ordered through ISIS

http://isis.uiowa.edu/ or by phone: call (319).335.0230 or toll free

(800)272-6430 and ask to be transferred.

http://registrar.uiowa.edu/transcripts/

http://isis.uiowa.edu/


A- 59


Self-Test #3

Work the following eight problems. Allow yourself two hours for

this self-test.

1. Find the cycle decomposition and order of the following permutation:

. =

589672413

987654321σ

Write σ as a product of transpositions.

2. Show that nn SA and that nA = ! n21

3. a. Let G and G′ be groups and let GG ′→:ϕ be a group

homomorphism. Show that ( )ϕK , the kernel of ϕ , is a normal

subgroup of G.

b. Let G be a group and GK . Find a group G′ and a

homomorphism GG ′→:ϕ ( ) KK =ϕwith .

4. Let { }Z∈+= babiaR , . Show that R is an integral domain.

5. Let G be an abelian group and let { }1somefor >=∈= meaGaT m .

Prove that T is a subgroup of G and that TG / has no element—other

than its identity element—of finite order.

6. Show that any group of order 4 or less is abelian.

7. a. Show by induction that 222 21 n+++ ( ) ( ).12161 ++ nnn =

b. Show that no integer 34 += nu can be written as 22 bau += where

a, b are integers.

8. Let R be a ring. Show that for aaRa ⋅==⋅∈ 000, .

22m050_sg

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