This is the official website of the course MAT347 at the University of Toronto in the academic year 2009-2010.
ANNOUNCEMENTS
Your project is due in December. Information about it is available here.
The first midterm took place on Friday, October 30. You can see the exam, sketches of the solutions and statistics here.
For the remainder of the course, the TA's office hours will be by appointment only.
Do you have any comments or complaints that you would like to submit to me anonymously? If so, you can do it here .
(If you do not mind doing it nonymously, talking in person or sending an email are good options.)
LOGISTICS
Instructor.
Alfonso Gracia-Saz ( alfonso (at-sign) math (dot) toronto (dot) edu )
Office: HU 1023. Go to 215 Huron St, take the elevator to the 9th floor and the stairs to the 10th floor. If the entrance door to the 10th floor is locked, knock. My office is the first one and I can hear the knocking from it.
Tel: 416-946-3771
Office hours:
Mondays 4-5, Wednesday 1-2, at HU 1023,
or by appointment (email or ask at the end of the class).
TA:
Zavosh Amir-Khosravi ( zak (at-sign) math (dot) toronto (dot) edu)
Office hours by appointment.
Tetbook. Dummit and Foote: Abstract algebra, Wiley. 3rd edition.
Class schedule:
Time: Mon 11-12, Wed 11-12, Fri 10-12. (There is no a priori separation between lecture and tutorial hours.)
Location: 229 RW.
For all other information, including midterm and exam details, see the course syllabus.
HOMEWORK ASSIGNMENTS AND CALENDAR
I encourage you to attempt the reading assignments before the lectures on that topic start.
I will post every homework assignment here at least one week before it is due. I will not update them without warning less than a week before they are due.
You are required to do all the problems in the homework set, but only the ones in bold and brackets are to be turned in on the day the homework set is due. Sometimes the not-to-be-handed-in problems will help you solve the to-be-handed-in problems. They are due at the beginning of the class. Late assignments will not be accepted.
PART 1: Basic theory of groups.
Wed Sep 9 -- Mon Sep 14 Wed Sep 16
Reading:
Section 0.1. - These are basic concepts that you should know perfectly before the course starts, and that I will not cover in lecture.
Section 0.2. - I will come back to cover this later in the course, but it will still be beneficial to read it now.
For those curious to learn more about how (computationally) hard it is to understand abstract groups given by generators and relations, you can read the first section of this paper. This may not make sense unless you have an interest in computer science or in logic and foundations.
Wed Sep 23 -- Mon Sep 28
Read sections 2.1, 2.2, 3.1, 3.2 (only till Corollary 9, not inclusive).
Read sections 3.3, 3.5, 4.1. You may want to reread sections 1.7 and 2.2. We will also cover Burnside lemma (problem 8 on page 877) and its applications.
This website may help visualize the group of symmetries of a dodecahedron or an icosahedron, and understand why they are isomorphic to A_5. Notice that you can rotate the images with your mouse. Here is another link.
If you have trouble visualizing the platonic solids, you can get a cheap set of dice that includes all five of them from any comic book store.
Homework #5 (due on Friday, October 16):
Section 2.2 (page 52): problems 6, 7, 10, [12].
Section 3.3 (page 100): problems 3, [7].
Section 3.5 (page 111): problems 3, 4, [7], 9.
Section 4.1 (page 116): problems 1, 6.
Also do the problems on this handout --Warning! Different handout -- and hand in problem [5]. (Some of these problems will be done in class.)
Microbes have interesting groups of symmetries (ignore the happy faces; the real things are mere blobs with protuberances). What are the groups of symmetries of the common cold virus, HIV, herpes, hepatitis, and mono?
On Friday Oct 23 we will do a mock test. I will give you one or two hard problems that could have been on the first midterm, and I will leave you (roughly) the first hour to work on them (alone or in groups, you may consult the book and your notes). During that hour I will also be available to answer individual questions. In the second hour, I will explain how to solve the hard problem(s).
(The problems we did are here.)
The first midterm will take place on Friday, October 30. More information here.
Problems 7, 8, 9, 10 in this handout.
Comments on the homework:
There is a typo in the book in problem 7 on section 6.3 (thanks, Trong!) One relation is missing. You can figure out which one.
Problem 6 in section 6.3 is useful to solve Problem 9 in section 6.3. If you are uncertain about your solution to Problem 6, feel free to check it with me.
Problem 7 in the handout is particularly important. For your project, you are likely to need to use it, or may even have to derive a variation of one of the results on problem 7 that is suitable
for your specific problem.
In order to solve Problem 10 in section 5.5 it will be useful to have some explanations which I will give in class on Friday, November 20. Hence I
have delayed the due date for this
assignment. The rest of the deadlines still stand.
Notes:
Here is a table with the number of non-isomorphic groups with order up to 2000.
As an example of a harder problem, here is the classification of groups of order 60 up to isomorphism. I recommend you postpone reading this until you are done with reading the sections on the textbook and doing all the homework.
We are postponing the proof of the Fundamental Theorem of Finite Abelian Groups till January.
The project is due in December. More information available here.
ADDENDUM: In some classifications, it is useful to be able to construct explicitly a matrix of a certain order of a certain size with coefficients on a certain field. (For instance, how do you construct a 3-by-3 matrix with coefficients in Z/2Z, and which has order 7? One approach to do this is to use the Normal Canonical Form theorem for matrices. This is usually covered in a first linear algebra course, but I have been told that some of you are unfamiliar with it. Hence, I have written some brief, sketchy notes about it here. Strictly speaking, you do not need this to solve your project problem, but it is a useful tool that sometimes provides shorcuts.
PART 3: Ring theory.
Mon Nov 23 -- Fri Dec 4
Read sections 7.1, 7.2, 7.3, 7.4.
On Friday, December 4 we will have a guest speaker: Dr Johanna Franklin, who is a logician, will give you an introduction to the Foundations of Mathematics. Specifically, she will explain what the Axiom of Choice and Zorn's Lemma are. We will be using Zorn's lemma often in the second term.
Homework #9 (due on Friday, January 8): TBA
WARNING: The rest of the schedule is tentative, and will be reviewed in January.
Mon Jan 4 -- Fri Jan 8
Read Sections 7.4, 7.5, 7.6, and Appendix I.
Homework #10 (due on Friday, January 15): TBA
Mon Jan 11 -- Fri Jan 22
Read Sections 8.1, 8.2, 8.3. During this period we will also do the proof of the Fundamental Theorem of Finite Abelian Groups, which we had pending.
Homework #11 (due on Friday, January 29): TBA
Fri Jan 22 -- Fri Jan 29
Read sections 9.1, 9.2, 9.3, 9.4, 9.5.
Homework: TBA
The second midterm is tentatively scheduled for Friday, February 5. The exact date and more information will be posted here.