Math 541, Spring 2009, Tonghai Yang

Lecture Section 1, MWF 1:20-2:10pm, B203 VAN VLECK



Text:  Abstract Algebra, by David Dummit and Richard Foote, Third Edition, John Wiley and Sons, Inc., 2004.
 

Office Hours: I will be in my office 705 Van Vleck from 10-11am on MF and 12:30-1pm on W. . These times may change, in which case the new times will be announced on my home page. Other good times to talk are right before or right after the class. It is also possible to ask questions by email, which will be generally answered within 24 hours. The more detailed your email question is, the more detailed the reply message will be. My email is thyang@math.wisc.edu.
 

Grading: The course grades will be computed as follows. There will be two midterms, the final and homework. Each midterm will be graded on a scale from 0 to 100, and the final will be graded on a scale from 0 to 150. The homework will be graded on the scale from 0 to 150 (each homework worths 15 points), with half-integers allowed. At the end of the semester, all these scores are added to give your total score, in the range from 0 to 500. The grades are given according to the total scores, and the average grade in the class is expected to be slightly above B. Improvement towards the end of the semester is not reflected in the final grade. Two people with the same total scores will receive the same grade, regardless of who did better at the end of the semester.
 

Midterm and Final Exams: We will have two 50 minute midterm exams. They will be given in class. There will be a two-hour final examination given on Wednesday May 11 at 10:05pm  in the location to be announced. You must take the final examination at the time scheduled by the university; no final exams will be given earlier. In particular, examinations will not be rescheduled because of travel arrangements. It is your responsibility to schedule travel appropriately. Notes/textbooks will not be allowed during the exams. Calculators will not be allowed, and will not be needed.
 

Homework: There will be eleven problem sets with five problems each. They are the suggested problems in the table below with *. . Each problem will be graded on the scale from 0 to 3. This is a proof-based course, which means that you will be expected to write rigorous proofs to get full credit. An essence of a rigorous proof is an air-tight logical argument that establishes the claim.  It is OK to talk to other students about homework problems. Most exam problems will be generally a subset of the suggested homework problems and the practice, but they will be worth a lot more points and will be thus graded in more detail.
 

Late/missed homework policy:  We don't  usually allow late homework unless a real good reason is given and accepted.   Homework turned in after the set has been handed out will score zero. . There is no make-up for homework. However, we will drop one of lowest scores from your HWs and thus counts 10 HWs with maximal 150 points.


Missed exam policy: There are no makeups for missed midterm exams, regardless of the reason for absence. However, if you can not attend the midterm due to a valid reason, for example a medical emergency, the rest of your exam scores will be scaled to compensate for the missed test. If you can not take the final exam due to a valid reason, you will be given the grade of Incomplete, with the makeup exam scheduled for the beginning of the Spring semester.
 

Miscellaneous: The goal of the homework sets is to introduce you to writing rigorously about standard mathematical objects. However, they will only scratch the surface when it comes to the mathematical properties of these objects, i.e. there are a lot more exercises in the book. If you plan on going to a decent PhD program in mathematics or theoretical physics, you should solve as many of the exercises as you have time for (though not necessarily write them down).




































 
                         Schedule of lectures and  Homework Problems:
Week
Sections
Topics
Suggested  HW problems
Jan. 19-23
Chapter 0, 1.1
Preliminaries, Definition of groups
0.1.4, 0.1.5*, 0.1.7*
0.2.1(a, b), 0.2.3, 0.2.8, 0.2.11*
0.3.4, 0.3.8, 0.3.9,  0.3.10, 0.3.15(b*, c)
1.1.1, 1.1.6*, 1.1.12, , 1.1.15, 1.1.21, 1.1.25, 1.1.28
Jan. 26-30
1.2-1.4
Dihedral groups, symmetric groups,  matrix groups
1.2.2*, 1.2.3, 1.2.9, 1.2.15, 1.2.17
1.3.2, 1.3.4, 1.3.5*, 1.3.7*, 1.3.11, 1.3.12, 1.3.15*, 1.3.16
1.4.1, 1.4.8*, 1.4.11
Feb. 2-6
1.5-1.7
Quaternion group, hom. and iso.,  group action
1.5.3*,
1.6.1*, 1.6.2, 1.6.7*, 1.6.16, 1.6.25
1.7.1, 1.7.8*, 1.7.13, 1.7.16*, 1.7.17, 1.7.18, 1.7.19,
Feb. 9-13
2.1-2.3
Subgroups,  Centralizers, normalizers, stabilizers, and kernels, cyclic groups/subgroups
2.1.2*, 2.1.6, 2.1.8, 2.1.9
2.2.2, 2.2.3*,  2.2.6*, 2.2.12,
2.3.3*, 2.3.8*, 2.3.10, 2.3.13, 2.3.16, 2.3.21
Feb. 16-20
2.4, 2.5, Review
subgroup gen. by a subset, lattice of subgroups,  review for Exam 1.
2.4.2, 2.4.6*, 2.4.8, 2.4.10*, 2.4.11, 2.4.14*, 2.4.16
2.5.3*, 2.5.4,  2.5.9(b)*,  2.5.19
3.1.1*, 3.1.5, 3.1.9, 3.1.11, 3.1.14*,  3.1.17, 3.1.19,3.1.32, 3.1.36
Feb. 23-27
Exam 1,  3.1
Exam 1 is on Monday or Wednesday, 
 Definition of quotient groups

Mar. 2-6
3.2-3.3

more on cosets, Lagrange's theorem, and the fundamental theorems of group homomorphisms
3.2.1, 3.2.6*, 3.2.11, 3.2.15, 3.2.18, 3.2.19, 3.2.22, 3.2.23
3.3.1, 3.3.3*, 3.3.7, 3.3.8*.


Mar. 9-13
3.5,  4.1,   4.3


$S_n$, $A_n$,   Conjugation action and the class equation
3.5.2, 3.5.5, 3.5.6*, 3.5.10*, 3.5.15
4.1.1, 4.1.4*, 4.1.10
4.3.2*, 4.3.9,  4.3.13*, 4.3.17, 4.3.20


Mar. 16-20
Spring Break


Mar.  23-27

4.4, 4.5
Automorphisms,  Sylow's theorems
4.4.1*, 4.4.3, 4.4.6, 4.4.13* (can use Section 4.5), 4.4.17, 4.4.18
4.5.5*, 4.5.8*, 4.5.14*, 4.5.26, 4.5.30

Mar. 30-Apr. 3
7.1, 7.2,  7.3
Rings and examples,  Ring homomorphisms and quotient  rings
7.1.1, 7.1.5*, 7.1.6, 7.1.10, 7.1.11*, 7.1.18, 7.1.26
7.2.1*, 7.2.3, 7.2.9*, 7.2.12
7.3.3*, 7.3.6, 7.3.10, 7.3.15, 7.3.19, 7.3.22
Apr. 6-10
7.3, 7.4, 7.5
Properties of ideals and  Ring of fractions
7.4.1, 7.4.4*, 7.4.9*, 7.4.14, 7.4.15, 7.4.17*, 7.4.20 7.4.30
7.5.2*, 7.5.3*, 7.5.4
Apr. 13-17
Review, Exam 2
 Exam 2 is on Wednesday or Friday.

Apr. 20-24
7.6, 8.1
Chinese Reminder Theorem, Euclidean Domain
7.6.1, 7.6.2*, 7.6.3, 7.6.5*, 7.6.6,
8.1.2(a)(b), 8.1.4*, 8.1.5*, 8.1.6*, 8.1.8, 8.1.12
Apr. 27-May 1
8.2-8.3
PID, UFDs
8.2.2*, 8.2.4*, 8.2.5*, 8.2.7, 8.2.8
8.3.1*, 8.3.3*, 8.3.5, 8.3.8
May 4-May 8
Review

10:05am-12:05pm
May 11
Final Exam