Math 546 (Algebraic Structures I) Section 001 Summer II

Lectures in BA 791

MTWThF 10:05 AM - 12:00 PM

Professor: Peter J. Nyikos Prof. Nyikos's Office: LeConte 406.     Phone: 7-5134 Office hours:MTWThF 8:30-9:55 or by appointment Grades are mostly determined by homework and quizzes (total score adjusted to 200 points), a midterm test worth 100 points, and the final exam, on August 2, worth 200 points. Attendance will be a factor in cases close to the borderline between two grades. The midterm for this course is on Monday, July 22. It covers Sections 1.1-1.4, Theorems 1.13 and 1.15, Theorems 2.1 and 2.2, Sections 2.2 and 3.2, converting from decimal notation and back, and the use of the Euclidean algorithm.

The textbook for this course is Abstract Algebra by W.E. Deskins. This Dover reprint is a classic and was chosen partly to save students money. It does have the drawback that some of the notation and terminology is obsolete, but there will be handouts that give the current notation and terminology when the need arises.

The course will cover part or all of the following sections:
1.1 through 1.5
2.1 and 2.2
3.1 through 3.5
5.1 and 5.2
6.1 through 6.7
7.1, 7.2, 7.4 and 7.6

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 second quiz in this course is on Wednesday, July 17. It covers the definitions in Section 1.4 and the proofs of Theorems 2.9 through 2.12 in Section 2.2.

Hand-in homework

First set of homework, due Wednesday, July 10.
1. Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} C = {1, 3, 5, 7} and D = {1, 3, 7, 9}. Find the following, where u stands for union and n for intersection:
A n C,   (A u C) n D,   (A u C) \ D,   and (A u C) n B.
2. Do (d) (e) and (f) in Problem 10, Section 1.
3. Do (d) (e) and (f) in Problem 11, Section 1.
4. Do (d) (e) and (f) in Problem 7, Section 1.2, in the following way: for each of the following properties, tell whether or not it holds in each case.
reflexive
symmetric
transitive
anti-symmetric
strictly anti-symmetric

Second set of homework, originally due July 15, but with the last problem now due Thursday, July 18.
Section 1.3, numbers 1, 3 [not counted for grade] and 5
Write out a table for the binary operation u (union) on the power set of {0, 1}
Section 1.4, number 1: write out the tables for all possible binary operations on {x, y} and tell which are commutative, and which of the commutative ones are associative.

Third set of homework, due Wednesday, July 17:.
Section 1.4, numbers 10, 11, 13
Tell which axioms for a Boolean algebra are NOT satisfied when E ={1, 2, 3} in Problem 8, page 30.

Fourth set of homework, due Thursday, July 18.
Finish Section 1.4, number 1 as above.
Section 3.1, number 1.

Fifth set of homework, due Tuesday, July 23.
Characterize the right zeroes in Example 3 on p.22 (see Problem 10 on the next page for definition).
Section 2.2, number 5;
Section 3.2, number 11;
Section 3.4, number 2; and
With [a]*b defined as [a^b] (where a and b are positive integers, and a^b means a to the bth power), write out the tables for [a]*b for the nonzero residue classes modulo 6 and modulo 7 and b > 0. You may omit brackets to simplify the display.

Sixth set of homework, due Tuesday, July 23.
Section 6.1, numbers 5 and 8
Section 6.2, number 1

Seventh set of homework, due Monday, July 29:
Section 6.3, numbers 5, 9, 15, 16
Section 6.4, number 5
Construct a homomorphism from the group of symmetries of an equilateral triangle (noncommutative group of six elements) to the additive (Z/(2), +) of residue classes modulo 2.
[Held over from Thursday] Write the tables for the commutative loops of order 5, and one noncommutative loop of order 5, using the numbers 1, 2, 3, 4, and 5 in that order, and letting 1 be the identity.

Extra Credit

Extra credit problems will be assigned from time to time. They are to be done strictly on your own, except that I am willing to give you advice. You are not to discuss them with anyone else.

There is no due date for extra credit, but once a fully correct solution is handed back, the problem is no longer eligible for extra credit. This is true of all problems with lines through them below.
If you can't quite get the solution to an extra credit problem but have some ideas, hand it in for partial credit. I will keep adding to your score as you improve your work on it.

1. p. 20, number 5 [6 points]

2. p. 20, number 12 [4 points each part]   you may use the definition of "inverse" that says F' is the inverse of F iff their composition F'oF is the identity on the domain of F.

3. Show that, in a Boolean algebra, (a+b)' = a'.b' and (a.b)' = a' + b'. [6 points] [The dot should be raised to denote "multiplication".

4. With (a - b) defined as (a'+b)' in a Boolean algebra, (a - b) + b = a+b. [6 points] [Hint: you may assume the preceding problem in solving this one.]

5. Characterize the right zeros for Example 3, page 22 [6 points].

6. Is there a commutative loop of 5 elements that is not a group? [3 points for correct answer, 12 points for a correct answer proven to be correct]

7. Write out the multiplication table for the quaternion group of Exercise 16, page 256 and find the subgroups of this group. [10 points]