MAT 312/AMS 351 Syllabus

Spring  2007

SUNY at Stony Brook
Department of Mathematics
SUNY at Stony Brook

This course is an introduction to number theory and to group theory, together with some applications to codes and coding theory.

Attendance: We reserve the right, at any time, to implement a policy of requiring lecture attendance. Such a policy, if instituted, will be announce on this website at least one week before it commences. During the period for which attendance is taken, anyone who misses lecture or recitation, shows up more than 10 minutes late, or leaves early, more than twice, will have their final grade reduced by one unit (e.g. from a B- to a C+).

Homework: Problem solving is a fundamental part of the course, and you are supposed to work hard on the homework assignments in order to succeed in the course. You are encouraged to discuss homework problems with other students. However, each student must write up the homework individually, in his/her words rather than merely copying someone else's. You will be required to turn in your homework assignment in recitation, following the week in which it was assigned. For example, the problems for the week 1/22-1/26 are due in recitation, during the week of 1/29-2/2. Late homework will not be accepted. No exceptions.

Warning about Calculators and Solution Manuals: Calculators and solution manuals can be of great assistance in helping you to learn the material, if used properly. If used improperly, they can actually cause great damage. Here is the proper way to use them, when you want to work on a problem:

First do the problem yourself, without touching the calculator or solution manual.

Then use the calculator or solution manual to check your work.

If the calculator or solution manual reveal any surprises, find a logical explanation for them.

Calculator abuse: When you first see a problem, your first response should be to think, not to punch buttons on a calculator; otherwise you are suffering from calculator abuse. Students with this syndrome lose out in the following ways:

They do not develop self-confidence in their own abilities to work the problems, which is essential for mathematical growth.

Mathematics is outside them, not part of them. You may have noticed that, if you write down a phone number, you are less likely to remember it. Similarly, calculator abusers often find themselves with poor memories for mathematics.

They do not learn to calculate well. Many courses in physics and the other sciences require students to be able to follow, and do, very complicated calculations.

In-class quizzes: There will be 3 short (15 minute) quizzes during the semester. These quizzes will be held in class on Wednesdays. The quizzes will be held on 2/7, 2/28 and 4/11.

Examinations: There will be two in-class midterm tests, on Wednesday, February 14, and Wednesday, March 21. The final exam will be on Wednesday, May 9 from 11:00 am to 1:30 pm. Make sure that you are available at these times, as there will be no make-ups for missed mid-term exams. Calculators, books, notes, etc. are not allowed during exams. If you miss an exam for an acceptable reason and provide an acceptable written excuse, the relevant mid-term will be dropped' in computing your course grade. A letter stating that you were seen by a doctor or other medical personnel is not an acceptable document. An acceptable document should state that it was reasonable/proper for you to seek medical attention and was medically necessary for you to miss the exam (the note/letter need not state anything beyond this point). Incomplete grades will be granted only if documented circumstances beyond your control prevent you from taking the final exam. You must have ID to be admitted to exams.

Grading: your course grade will be based on your examination performance, quizzes, and homework, weighted as follows:

Midterm I 20%
Midterm II 20% 
 Final Exam 40%
 Quizzes  10%
 Homework  10%

Help: The Math Learning Center (MLC) is in S-240A of the math building. This is a place where students can go for help with pre-calculus and calculus material and where study groups can meet. The MLC is open 11am-9pm Monday through Wednesday,   11am-6pm Thursday and 11am-2pm on Friday. For more information on the Math Learning Center, please click here.

DSS advisory: If you have a physical, psychiatric, medical, or learning disability that could adversely affect your ability to carry out assigned course work, we urge you to contact the staff in the Disabled Student Services office (DSS), Room 133 Humanities, 632-6748/TDD. DSS will review your situation and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential.

Schedule (tentative): The following is the basic syllabus, but not all topics in each section will get covered. Please read the relevant parts of the book before class.

Week Sections Notes Homework
1/22-1/26 1.1,1.2 Office hours this week: MF 12:45-2:15  1.1:#1(iii)(v)(show all work), 3, 6.
Also prove that the least common multiple of any two positive integers a, b exists and equals ab/d, where d=(a,b). (See the definition on page 14.)
Bonus problem: A frictionless rectangular billiard table is a feet by b feet, where a and b are integers, and it has pockets only at the corners. A ball is shot at a 45 degree angle onto the table from one of the corner pockets. Collisions with the sides of the table are completely elastic (that is, the ball doesn't lose any of its speed when it hits a side of the table and bounces off). Show that the ball will eventually go into one of the other pockets, and determine exactly how far it travels.
1/29-2/2 1.3,1.4 Office hours this week: MF 12:45-2:15.
Last day to add or drop a course without a W: February 2 
1.3: 2, 5, 6, 8.
Hint for #6: say n is at least 4; and for instance, why can't n be even?
1.4:1(i)(ii)(iii)(v), 2 (they are asking for the addition and multiplication tables mod n. Do n=4 and n=5 instead of n=6 and n=7 since those are in the back of the book), 5 (see the argument at the bottom of page 37 for a related problem), 9(ii).
Also: A professional poker player actually performed the following trick on ESPN2: someone showed him 51 cards of a deck, one at a time, and he was then immediately able to name the last card. Explain how you, too, could perform this trick.
2/5-2/9 1.5,1.6 First quiz: Wednesday, February 7, on sections 1.1-1.3.  1.4:6, 7.
Also: find the inverse of 9 mod 11; the inverse of 13 mod 31; and the inverse of 89 mod 237.
1.5:1(1)(ii)(iii)(v)(vii)m 2(i)(iii), 32(i)(iii), 3
The answers for all these problems are in the back of the book, so you have to show work if you want any credit.
1.6: Find the orders of 3 mod 80 and 2 mod 21.
Also: 2(i)(iii). (The answers for all these problems are in the back of the book, so you have to show work if you want any credit.)
2/12-2/16 review, finish 1.6 First midterm: Wednesday, February 14, through section 1.5
Approximate grade equivalents: (A: 80-100)(B: 70-79)(C: 50-69) (D: 30-49) (F: 0-29)
Note that, here, for instance, the "B" range includes B+ and B-. 
1.6: 3, 5 (show work), 6(i)(ii) (show work), 7, 10 (show work), 12 (show work)
2/19-2/23 2.1,2.2   2.1: 2, 4, 7
2.2: 2(ii), (iii), (v) (justify all answers), 6.
Also: show that every increasing function from the reals to the reals is injective.
2/26-3/2 2.3 Second quiz: Wednesday, Feb. 28, covering section 2.2   2.3: 1(a)(c)(e)(g)(prove all your assertions; if a property doesn't hold, give a counterexample); 2(a)(b); 7 (give complete explanations), 8 (give complete explanations).
3/5-3/9   4.1: 1 (first six products), 2 (first 3 inverses).
Also write each of the five permutations pi_1, and pi_2 and ... and pi_5 on page 158 as a product of disjoint cycles.
Also compute (1234)(1632)(2463)(1542) as a product of disjoint cycles.
4.2: 1(i)(iii)(v) (show work), 4, 7 (show work), 11 (show work), 13.
3/12-3/16 4.3   4.3:1(v)(vi)(vii) (if it's a group, prove it; if it isn't, give a DETAILED counterexample to one of the group properties), 2, 3, 4 (also explain why 1(viii) is a special case of problem 4), 5, 8
3/19-3/23 review, finish 4.3 Second midterm: Wednesday, March 21, through section 4.3.

We haven't covered and you aren't responsible for:
Section 4.3.4 (pages 177-183);
Pages 107-111 and example 4 on page 112 in section 2.3;
Pages 98-100 in section 2.2;
Any math history.

Last day to P/NC or drop a course: March 23
Homework: page 211, #1.
Also: In a group G, find x if axb = c.
Also: Let G be an abelian group with m elements. Show that for any g in G, gm is the identity. (Here as usual, g2 = gg, g3 = ggg, etc.)(Hint: imitate the proof of Theorem 1.6.7.)
Note: in these exercises, you're not allowed to choose your own example of a group!! Your arguments should work no matter what the group is.
These problems might be easier after the class on Monday 3/26, but try to do think about them over the weekend too.
3/26-3/30 5.1   Homework: pages 211-212: #4 (if it is a subgroup, prove it; if not, give an explicit counterexample to one of the required properties), and also #9.
Also do #10, but since part of the answer is in the back of the book, find generators for G17 and G19 instead of G23 and G26; also show G8 is not cyclic (that is, there is no element which generates it).
Also: find all subgroups of Z6, the group of congruence classes mod 6 with addition as the operation.
4/2-4/6 Spring break; no classes in session 
4/9-4/13 start 5.2, do 4.4 Third quiz: Wednesday, April 11, on Section 5.1  Homework: Consider the integers Z as a group under addition. Let H be the subset of Z consisting of all multiples of 5. Show that H is a subgroup, and write down all of its distinct left cosets.
The group S(4) is the group of permutations of the set {1,2,3,4} (see page 174). Let H denote the subgroup of S(4) generated by the cycle (1234) (see page 209). Thus H=<(1234)>. Write down the elements of H. Also write down all distinct left cosets of H (see the top of page 214 for a related example).
Section 4.4: 1(i)(ii)(iii) (prove your assertions), 3(i)(ii)(iv)(v) (prove your assertions; in (iv) Venn diagrams are OK), 5, 9 (prove your assertions), 10(i)(ii). In these problems, you may assume it is known that addition or multiplication, of numbers or matrices, is associative.
4/16-4/20 finish 5.2, do 5.3   Homework: 5.2: 2, 4;
5.3: 1 (prove your assertions), 4 (prove your asertions), 5.
4/23-4/27 finish 5.3, 5.4  
4/30-5/4 finish 5.4; overview of chapter 6; review Last day of classes: May 4  Homework: Due Friday, May 4: #2 (prove your assertions), 7 (use a parity-check matrix)
5/7-5/11 No classes Final Exam Wednesday, May 9, 11:00 am -1:30 pm., in our regular classroom. Covers the whole semester's work, with emphasis on material covered since the last midterm.
We haven't covered and you aren't responsible for:
Section 4.3.4 (pages 177-183);
Pages 107-111 and example 4 on page 112 in section 2.3;
Pages 98-100 in section 2.2;
Section 2.4;
Chapter 3;
Coset decoding tables (pages 242-243);
Chapter 6;
Any math history.

The practice exam from Fall 2006 is here, and the solutions are here, Let me stress that another instructor made up this practice exam for his own students, so our exam may be quite different from this one. Note also that, in problem 16, we never discussed two-column decoding tables. In general, if you go over the homework problems, midterms and quizzes from this semester's course, that would be a far more reliable indicator of what might be on our exam than this practice exam is.
Would you like to be a teaching assistant for a MAT course in the fall?
If you are doing well in math, please consider being one. It looks great on your resume. Come to the Math Undergraduate Office (P-143 Math Building) and fill out the form.

Undergrads teach sections of the remedial courses MAP 101 and MAP 103, under the supervision of the faculty coordinator for those courses. They also do recitations for MAT 123 (precalculus) and occasionally for other courses as well (118, 122, 125).

For the first course you do for us, you are given supervision, and credit for the course MAT 475 (Undergrad Teaching Practicum); you don't have to pay tuition for MAT 475. We pay you for all further courses that you teach for us, whether in the same semester as the first course or in later semesters.

Please be aware that if you want to teach in the spring but not in the fall, it's not likely you will be hired. Our teaching needs in the fall are vastly greater than in the spring. Usually we rehire fall people for the spring and there are no additional positions.