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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Continuing Education Division of Continuing Education
250 Continuing Education Facility Iowa City, IA 52242-0907
Telephone: 319-335-2575 • Toll free: 1-800-272-6430
Fax: 319-335-2740 • E-mail: [email protected] Web: http://continuetolearn.uiowa.edu/ccp/
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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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.
Continuing Education The University of Iowa
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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.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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
Continuing Education The University of Iowa
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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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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
Continuing Education The University of Iowa
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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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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
Continuing Education The University of Iowa
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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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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
Continuing Education The University of Iowa
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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
Reading Assignment and Comments
• 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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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.
Written Assignment #2
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 13: 2, 7, 14, 16, 17, 29.
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Lesson 3 A(S)—The Set of Bijections on S
Reading Assignment and Comments
• Read Herstein: Section 1.4, pages 16–18.
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.
Written Assignment #3
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.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Description
• Page 19: 3, 5, 10, 17, 23.
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Lesson 4 The Integers
Reading Assignment and Comments
• 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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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,
Continuing Education The University of Iowa
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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.
Written Assignment #4
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 28: 1, 6, 8, 11, 13, 14.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 5 Mathematical Induction
Reading Assignment and Comments
• 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.
Written Assignment #5
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.
Continuing Education The University of Iowa
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Description
• Page 31: 2, 8, 10, 13.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 6 Complex Numbers
Reading Assignment and Comments
• Read Herstein: Section 1.7, pages 32–37.
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,
Continuing Education The University of Iowa
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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
2010 sincos2 ππππ
( ) ( )( ) andsincos2 56
2056
2010 , + i + + ππππ
( ) ( )( ) . + i + + 58
2058
2010 sincos2 ππππ
Practice Exercises
• Page 37: 1, 5, 11, 20.
Written Assignment #6
Instructions
Assignments can be submitted in print or via e-mail. See your
course syllabus (provided with print Study Guides and available on the
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Description
• Page 37: 2, 4, 13, 14, 22.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Continuing Education The University of Iowa
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Lesson 7 Definitions and Examples of Groups
Reading Assignment and Comments
• Read Herstein: Section 2.1, pages 40–46.
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:
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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
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 46: 3, 4, 15, 17, 25, 30.
Continuing Education The University of Iowa
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MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 8 Some Simple Properties of Groups
Reading Assignment and Comments
• Read Herstein: Section 2.2, pages 48–50.
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.
Continuing Education The University of Iowa
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Written Assignment #8
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 50: 2, 4, 6.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 9 Self-Test #1
Reading Assignment and Comments
• 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.
Written Assignment #9
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
• 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.
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
Continuing Education The University of Iowa
A- 33
Examination Request Form located at the back of this Study Guide (print
version only).
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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.
Continuing Education The University of Iowa
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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
Lesson 15 Self-Test #2
Examination #2
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 10 Subgroups
Reading Assignment and Comments
• Read Herstein: Section 2.3, pages 51–56.
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.
Continuing Education The University of Iowa
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Written Assignment #10
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 54: 1, 8, 10, 12, 15, 17, 20. (Hint: Try 3SG = ).
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
A- 38
Lesson 11 Lagrange's Theorem
Reading Assignment and Comments
• Read Herstein: Section 2.4, pages 56–63.
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
Continuing Education The University of Iowa
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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.
Written Assignment #11
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 63: 1, 4, 5, 6, 9, 22, 25, 31.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 12 Homomorphisms and Normal Subgroups
Reading Assignment and Comments
• Read Herstein: Section 2.5, pages 66–73.
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
Continuing Education The University of Iowa
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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.
Written Assignment #12
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 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.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 13 Factor Groups
Reading Assignment and Comments
• 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
Continuing Education The University of Iowa
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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.
Written Assignment #13
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 82: 1, 3, 8, 9, 12, 18.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 14 The Homomorphism Theorems
Reading Assignment and Comments
• Read Herstein: Section 2.7, pages 84–87.
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.
Continuing Education The University of Iowa
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Practice Exercises
• Page 87: 1, 2, 3.
Written Assignment #14
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 87: 4, 6, 7.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 15 Self-Test #2
Reading Assignment and Comments
• Review the material in Sections 2.3 through 2.7 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 your answers and send your
answers to your instructor for grading.
Written Assignment #15
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
• Complete the self-test which follows.
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.
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
Continuing Education The University of Iowa
A- 47
Examination Request Form located at the back of this Study Guide (print
version only).
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
A- 48
22M:050 Introduction to Abstract Algebra I
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.
Continuing Education The University of Iowa
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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
Lesson 20 Self-Test #3
Final Examination
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 16 Permutations and Cycles
Reading Assignment and Comments
• 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).)
Continuing Education The University of Iowa
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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.
Written Assignment #16
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.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Description
• Page 110: 2, 3, 5.
• Page 117: 1, 3, 5, 8, 20.
Continuing Education The University of Iowa
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Lesson 17 Odd and Even Permutations
Reading Assignment and Comments
• Read Herstein: Section 3.3, pages 119–123.
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.
Written Assignment #17
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 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 !)
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 18 Rings I
Reading Assignment and Comments
• Read Herstein: Section 4.1, pages 125–133.
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
Continuing Education The University of Iowa
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Practice Exercises
• Page 133: 1, 2, 4, 10, 13, 20.
Written Assignment #18
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 133: 3, 5, 7, 16, 19.
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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Lesson 19 Rings II
Reading Assignment and Comments
• 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.
Written Assignment #19
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 139: 1, 2, 3, 6.
Continuing Education The University of Iowa
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Lesson 20 Self-Test #3
Reading Assignment and Comments
• Review the material in Chapters 1, 2 (Sections 1–7), 3, and 4
(Sections 1–2) 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.
Written Assignment #20
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
• Complete the self-test which follows.
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.
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
MATH:3720:0EXZ (22M:050) Introduction to Abstract Algebra I
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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).
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.
Continuing Education The University of Iowa
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22M:050 Introduction to Abstract Algebra I
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, .