MATH 417(317) B1U, Fall 2004. Introduction to Abstract Algebra

Check your current grade for this class.
Please see the correct date and time of exams below.

Professor: Igor Mineyev, 243 Illini Hall. mineyev@math.uiuc.edu
I would recommend talking to me during and after the class and coming for office hours, rather than sending an email. Email does not seem to be very efficient for discussions.

Office hours. Monday and Wednesday, 4:00-5:00 pm in Illini Hall, room 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.

Text. Algebra Pure and Applied, by A. Papantonopoulou, Prentice Hall, first edition, 2002.
Class time. MWF 9:00a.m.-9:50a.m.
Class location. 143 Henry Bld

Prerequisites. MATH 315 Linear Transformations and Matrices and either MATH 247 Fundamental Mathematics or MATH 248 Fundamental Mathematics - ACP.

Times of exams. Exam 1: Friday, October 1, 2004.
Exam 2: Friday, November 12, 2004.
Final Exam: 8:00 a.m.- 11 a.m., Thursday, December 16, 2004.

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 out the completely anonymous online evaluation form. (The form should be working now.) Your feedback will be used to improve teaching effectiveness 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. 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.

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.

Grading policy. If you have specific questions about the grading of your work, please see Mehmet Sahin on Tuesday between 1.30p.m. and 2.00p.m. in Coble Hall, room B1A.

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.

Homework and exams. It is important to learn the material and to do the homework after each class. 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.

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

How to organize homework, quizzes, and exams.

• Please write the problems in the order they were given.
• Staple the 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 417 B1U", "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 "n+1(n+1)=(n+1)²".
  
• 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.
• "<=>" means "if and only if". Never use this symbol to mean something else.
• Equations can have solutions. Functions cannot have solutions.
• If S is a number, then S cannot be true or false.
• Never copy any assignments. It is acceptable to discuss the material with others. It is not acceptable to copy homework.

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, this will most likely affect your grade. Abstraction is an impotrtant skill, it is the best and the most important part of the course, and it is your main goal to work on.

You must desire to work with abstract concepts. If not, this course is a wrong choice for you.

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.

It is your responsibility to know at what point you stop understanding, and to ask questions based on what you understand about what you do not understand. 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 important additional comment: Please make sure to have enough sleep before the class. 9 hours, no less. It helps better understanding. Exercise, jogging, and/or cold shower in the morning does not hurt either.
• If you need additional feedback on your homework, quizzes and exams, please ask me during the class and come for office hours.
• If you wnat me to solve additional prolems in class, I am more than willing to do that. It is your responsibility to ask for it.
• Check your current grade for this course. There should be a grade displayed for each homework, quiz and exam, and also a total grade based on the average score of your quizzes, homework and exams you have taken so far. Let me know if you cannot login or to see the grades.
• The curve for the quizzes, homework, and overall:
  A+- 82-100, B+- 72-82, C+- 62-72, D+- 52-62, F<52
  The curve for Exam 1: A+- 77-100, B+- 67-77, C+- 57-67, D+- 47-57, F < 47.
  The curve for Exam 2: A+- 82, B+- 72, C+- 62, D+- 52, F<52.
  For Exam 1, the median was 71.5%, mean 68.6%, standard deviation 21.1%.
  For Exam 2, the median was 82%, mean 76.7%, standard deviation 23.7%.

  How to read your grade records: Note the overall curve above. If your record says ex1:53< 58/100, this means that you received 53 points for the first exam, and it was rescaled to 58 because of the curves used. 58 is in the range 52-62 for D+- in the overall curve. This means that your rescaled score 58 gives letter D for the first exam. Please pay attention to the letters, not to the numbers.

  The same is for the average: if your average is 66, it might look like a small number, but it corresponds to C in the overall curve.

  ** denotes the dropped lowest quizzes and homework assignments (two each).

• LAS and Engineering students enrolled in this course are permitted to drop from this course without academic penalty and without petition through the end of the 13th week of the semester. Thus, Friday, November 19, 2004, is the drop deadline in Fall 2004.

  LAS students who wish to drop the course should see the receptionist in the LAS office and ask to be directed to a COAR user, who will process the late drop without a grade "W". Engineering students should go to the Engineering College Office, 206 Engineering Hall, for assistance with the late drop. Students in other colleges may need to petition their college office for this policy to be applied, but the math department will support it.

  Any request to drop the class after November 19, 2004, will be considered a late drop request, and the student will be required to submit a written petition to the Late Add/Drop Committee consistent with normal policy.

• Homework.

Beyond the course

• 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.