Math 546 (Algebraic Structures I) Section 001
Summer II, 2005

Lectures in LeConte 310

MTWThF 10:05 AM - 12:05 PM

Professor: Peter J. Nyikos

Prof. Nyikos's Office: LeConte 406.     Phone: 7-5134

Usual office hours: MTWThF 8:55-9:55 and MTWTh 12:30 - 1:30 pm or by appointment (or any time I am in). Exceptions posted on door and announced in advance whenever possible.

Special office hours July 30 and 31 (Thursday and Friday): 9:30 - 10:30, 11:30 - 12:30 and 1:00 - 2:00. In addition, there will be help sessions in LeConte 310 (your usual classroom) both days, from 10:30 to 11:30

Grades are mostly determined by homework and quizzes (total score adjusted to 200 points), a midterm test worth 100 points, and the final exam, worth 200 points. Attendance will be a factor as outlined here, where you can find other information on policies and grading, except that the final exam is 40% of your numerical grade, as is your combined homework and quiz grade, with the midterm counting for 20%; and there may be more than 3 homework assignments.

The midterm test was on Monday, July 20, and covered everything covered in class up through Wednesday, July 15.

The final exam is on Friday, August 1.

The last day to withdraw with a W was Friday, July 17.


The textbook for this course is Contemporary Abstract Algebra, Sixth Edition, by Joseph A. Gallian, ISBN: 0-618-51471-6.
The course covered the highlights of each chapter from 0 through 10, and also of chapters 12. Excerpts from chapters 11 and 13 were also covered. The objectives of this course are to attain a strong understanding of the basics of group theory and ring theory, with enough background in set theory to understand such basic concepts as sets, relations (including equivalence relations and partial order relations) and functions, and enough background in number theory to provide useful examples.

The first quiz was on Wednesday, July 8, with a problem very similar to Exercise 1 on page 23.
The second quiz was on Monday, July 13 on a problem or problems very similar to a practice exercise (see below) from Chapter 1 and/or Chapter 2.
The third quiz was on Friday, July 20 on problems very similar to a practice exercise (see below) from Chapter 3 and/or Chapter 4.

Practice exercises, not to be handed in:
Chapter 0: 1, 5, 9, 16
Chapter 1: 2, 5, 11, 13, and the first row of 22.
Chapter 2: 1, 2, 3, 5, 8, 19, 23.
Chapter 3: 1, 11, 31, 33, 35, 38, 47, 49.
Chapter 4: 1, 2, 3, 7, 8, 13, 17, 23, 27, 38, 43.

If you get stuck on any odd-numbered problem, see if you can understand the answer in the back.


First homework, due on Thursday, July 9:
1. Using the Euclidean Algorithm, find gcd(482, -288) and express it as a linear combination of 482 and -288.
2. Find the GCD of 482, -288, and 204. Hint: this will be gcd(n, 204) where n = gcd(482, -288).

Starred homework problems mean you have more than one chance to get credit. A single star means half credit for any improvement of the second attempt on the first attempt, while a double star means full credit for that and half credit for any improvement of the third attempt over the second attempt.

Second homework, due Thursday, July 16:
1. Do a Venn diagram of the subgroups of D_4, the group of symmetries of a square (Chapter 1). [Hint: Lagrange's theorem (The order of a subgroup divides the order of a group) is of help in making sure you have them all.]
*2. Show that the answer to 51 (b) [6th edition] and 59 (b) [7th edition] on page 71 is correct.
3. Write out the Cayley tables for Z_2 x Z_2 and Z_2 x Z_4. Here x denotes Cartesian product, with addition done coordinatewise, and Z_n stands for the group of integers modulo n, with the operation of addition.
*4. Show that U(8) is isomorphic to Z_2 x Z_2 and U(20) is isomorphic to Z_2 x Z_4. [Hint: the Venn diagram of subgroups of these groups is helpful.]
**5. Let S be a set with a binary operation *. Show that if (S, *) is isomorphic to a group, then (S, *) is a group.

The third set of homework was on a sheet that was distributed at the end of the midterm.

Homework 4, due Monday July 27
1. Problem 52 (6th ed.) = Problem 54 (7th ed.) of Chapter 5
2. Problem 10 of Chapter 6
*3. Problem 22 (6th ed.) = Problem 24 (7th ed.) of Chapter 6
4. Problem 10 of Chapter 7
5. Problem 22 (6th ed.) = Problem 24 (7th ed.) of Chapter 7
**6. Problem 36 (6th ed.) = Problem 40 (7th ed.) of Chapter 7

Homework 5, due Tuesday July 28
*1. Let Q_8 denote the quaternion group. [In the notation of the handout, it is the multiplicative group of the eight quaternions 1, -1, i, -i, j, -j, k, -k.]
  (a) Draw the Venn diagram of subgroups of Q_8.
  (b) Show that every subgroup of Q_8 is normal.
2. Problem 10 of Chapter 9
3. Problem 10 of Chapter 10
4. Problem 20 of Chapter 10
5. Problem 6 of Chapter 12
*6. Problem 8 of Chapter 12