Math 317 F1, Spring 2003

MATH 317 F1, Spring 2003. Introduction to Abstract Algebra

Read the information below completely. More information will be posted here as we proceed with the course.

Professor: Igor Mineyev, 235 Illini Hall.
mineyev@math.uiuc.edu Generally, I would recommend talking to me during and after the class and coming for office hours, rather that sending an email. Email does not seem to be very efficient for discussions.

Office hours. Monday and Wednesday, 3:00-4:00 pm in IH 235. I am very much available at other times if you cannot come for the regular office hours and you make an appointment with me after a class.

Grader. Xinyun Zhu, B1 Coble Hall.
If you have specific questions concerning grading, please see the grader on Friday from 3:40 to 4:10 pm. If you are convinced that your grade should be different, please write your name and a short paragraph on a separate piece of paper describing your reasons, and pass it to the grader.

Text. Algebra Pure and Applied, by A. Papantonopoulou, Prentice Hall, 2002.
Class time. MWF 2:00pm-2:50pm
Class location. 142 Henry Bld

Prerequisites. MATH 315 Linear Transformations and Matrices and either MATH 247 Fundamental Mathematics or MATH 248 Fundamental Mathematics (Advanced Composition).

Feedback from you. I am trying hard to maximize teaching effectiveness and the overall quality of this course.

If you have comments, sugestions, criticism etc for this course or for its instruction, you are very welcome to fill in the completely anonymous online evaluation form. Your feedback will be used to improve teaching efictiveness and the overall quality of the course.

Fedback on your course progress through homework, quizzes, and tests. Please pay a lot of attention to the marks and notes on your homework, quizzes, and tests that you receive from the grader. This is the important feedback on your course progress. If something is circled or underlined, it means that that part of your work contains a mistake or is not properly explained. The question mark sign, "?", usually means that you have not provided adequate explanation in that particular place. Make sure to go over each homework, quiz, and test, until you know what was wrong and how to fix it.

In addition to the above, I provide feedback on your course progress by

answering your questions during the class,
discussing problems from homework, quizzes, and tests,
providing solutions to exams and selected homework problems,
communication with you after the class and during my office hours.

You are strongly encouraged to ask questions. Asking questions is an important skill and an indispensable tool for obtaining feedback on your progress.

The course. This course brings you to the next level of sophistication from what you saw in the prerequisite courses. The goal of the course is to let you learn several abstract algebraic concepts, rigorous proofs of some important theorems, and their applications.

Tentative schedule.

The integers. Division algorithm. Greatest common divisors. Euclidean algorithm. Congruence arithmetic.
Permutations. Cycle decomposition. Order of a permutation. Even and odd permutations.
Groups. Definition and examples. Subgroups, cosets and Lagrange's Theorem. Euler's theorem. Applications to RSA cryptography and Key Exchange Protocol. Normal subgroups and quotient groups. Homomorphisms. The Isomorphism Theorems with illustrations.
Group actions. Examples of group actions. Orbits and stabilizers. Cayley's theorem. Burnside's theorem. Applications to the class equation, centers of finite p-groups, Cauchy's theorem. Simplicity of A₅. Applications to Polya counting theory.
Rings. Definitions and examples. Polynomial rings. Subrings and ideals. Quotient rings, homomorphisms. Isomorphism Theorems for rings. Integral domains and fields. Field of fractions of a domain. Division algorithm for F[x], F a field. Roots of polynomials. The remainder theorem.

The above schedule is tentative. Some adjustments might be made during the course.

Homework and exams. The homework will be assigned daily and collected weekly. 15-minute quizzes will be given every week. There will be two 50-minute midterm exams and a comprehensive final exam from 1:30pm to 4:30pm, Monday, May 12, in the regular classroom, as found in the Timetable. The time of the final exam will not be changed. Please make your travel arrangements so that not to miss the final exam. The quizzes and exams will not be necessarily the same as in the homework, but rather will be based on the homework and the material presented in class. No textbooks, lecture notes or calculators will be allowed on the exams and quizzes. No late assignments will be accepted. Missed or late assignments, exams, and quizzes count as 0.

Grading policy. Two lowest homework assignments and two lowest quizzes will be dropped. The grading scale will be decided individually for each exam after the scores are obtained, but it is guaranteed that you get at least A- if the score is 90%, at least B- if the score is 80%, at least C- if the score is 70%, and at least D- is the score is 60%. Each of the midterms will count as 17% of the grade, the homework 17%, the quizzes 17%, and the final exam 32%.

The maximal score for each homework and quiz is 10. For example, if you received 5 on a quiz, it corresponds to 50%. The grades will reflect your accomplishment. You can check your grades available so far.

Homework and exams policy. It is important to learn the material and to do the homework after each class, even though it will be collected weekly. Please write the problems in the order they were given. Staple the sheets. (No folding over corners.) Put "Last name, first name", "MATH 317 F1", "HW#". You are expected to write complete, grammatical sentences, and to spell correctly. Copying homework is unacceptable. There will be no "extra credit" or "make-ups". The reason two quizzes and two homework assignments are dropped is to allow for personal emergency situations, when there is a strong reason you must miss a class. Do not miss exams, quizzes and regular classes.

Attendance. It is very important to attend the class. If you miss a class, you are still responsible for knowing the material what was discussed. This includes the homework assigned during that class as well.

See the homework.

Notes on how to organize homework, quizzes, and exams.

Staple the weekly homework together, to make sure no part of it is lost. (No folding over corners.)
PRINT your "Last Name, First Name" legibly on the top of the homework. Also write "MATH 317 F1", "HW#" and the number. Write the solutions to the problems in the order they were assigned. Write the section number, then problem numbers.
Turn the homework at the beginning of the class it is due.
Always write complete solutions. Correct anwers with no solution do not score points.
If asked to answer a question, write the solution and the answer. If the question is to prove a statement, write a complete proof.
Use the equality sign, "=". Quite often, when you need to show that one thing is equal to another, you write a sequence of equalities connecting the two things. This is the best way to present a solution, whenever possible.
Use the equality sign correctly. The thing on the left side of "=" must actually be equal to the thing on the right side of "=".
Do not write statements that are obviously wrong, for example "P(n)=0<1", "n+1(n+1)=(n+1)²" or "f(g)=f(g(x))". (Why wrong?)
When you explain things by words, write complete grammatical sentences. It is wrong to write "Therefore, 5+3."
Writing correct things increases your grade. Writing nonsense reduces your grade.
"=>" means "implies". Never use this symbol to mean something else.
Equations can have solutions. Functions cannot have solutions.
Never copy any assignments. It is acceptable to discuss the material with others. It is not acceptable to have similar homework solutions.

On a philosophical note. You might be already familiar with the observations below, from your past experience with math courses. Nevertheless, I feel obliged to provide some general comments to make sure we all understand the expectations.

This is a course in abstract algebra. Its main goal is to help you learn abstract concepts and to further develop your abstract thinking. If you are unwilling to think in abstract terms (or in general, rather than in particular) or to put effort and time in order to achieve that state, your grade will most likely be affected.

If you would like to have an easy high grade, this course is not the right choice. If you like challenge, are ready to take merit when you succeed and to accept responsibility in case of failure, this is your course.

Just learning how to do problems is not enough. This is certainly necessary, but not sufficient. You really need to make sure that you will be able to do any similar problem at any other time, and without looking in the book. Problems in the tests and quizzes will be based on the knowledge you obtain from homework and the material discussed in class. You really need to know both.

In order to get used to abstract concepts, time and effort is required. It is important to work regularly because:

it is impossible to learn all the material several days before an exam;
if you have not learned some concepts and proofs properly at some time during the course, there is a good chance you will be lost for the rest of the course.

I will be more than happy to help you during the class and my office hours. And you will need to ask questions. If you don't, there is no way for me to figure out what kind of help is needed. No one is able to answer questions that are not asked.

It is your responsibility to know at what point you stop understanding, and to ask questions based on what you know about what you do not know. If at some point you realize that you have been lost for a whole week, this is because you did not ask a question a week ago.

It is my responsibility to cover all the material, and in order to do that, I must proceed at certain pace. It is your responsibility to follow at about the same pace, and if more time is required, once again, I am more than happy to provide help during my office hours. It is impossible to cover the whole material of the course during the last office hour before the exam. Come for office hours regularly during the semester, not only just before the exam.

It is also a good idea to discuss the material with other students after class hours, and to seek additional help elsewhere, if needed. Simply copying your friends' homework is a trap: it leaves you under a false impression that you know the material, and results in a big surprise after the exam.

Consistent effort will be needed through the course. Working slightly ahead of schedule might be helpful, if you so choose.

Additional notes

The first midterm exam was on Friday, February 28. The average (mean) for this exam was 75.3%. The following scale will be used for this exam:
90-100 is A's, (that is A- is 90-93.33, A is 93.33-96.66, A+ is 96.66-100), 80-90 is B's, 70-80 is C's, 60-70 is D's. These are only estimates. The overall grade will be based on your average score according to the percentages described earlier.
The second midterm exam was on Friday, April 4. The mean was 65.4. The scale: 85-100 A's, 70-85 B's, 55-70 C's, 40-55 D's.

You can check your grades. The average, based on the grades available so far and taking into account the the weights for homework, quizzes and exams, is automatically calculated. "4>5/10" means that your actual score was 4 but it was rescaled to 5 because of a different curve used; and the perfect score was 10. This is to give you an additional feedback on your progress.

Beyond the course

The following material is quite beyond the MATH 317 course, but if you are interested to see what group theory looks like, see for example,

open problems in combinatorial and geometric group theory or
the problem list (.dvi file) maintained by Mladen Bestvina, University of Utah.

You might also try to work on one of the seven Millennium Prize Problems, for example, the Poincare conjecture. Working on either one of these seven problems will make you either rich or very frustrated, depending on your ability, persistence, and luck.