Holy Cross Mathematics and Computer Science

Mathematics 351 -- Modern Algebra I

Syllabus Fall 2018

Professor: John Little
Office: Swords 331
Office Phone: 793-2274
Email: jlittle@holycross.edu
Office Hours: M 2-4pm, T 1-3pm, W 10-10:50am, R 10-12noon, F 1-2pm, and by appointment

Course Description

Algebra is the part of mathematics that arose historically from the problem of solving equations of various forms. A typical example here is the problem of solving quadratic equations a x² + b x + c = 0. The well-known quadratic formula from high school mathematics gives a general way to do this, of course. Although the symbols we use here are relatively recent (from the 17th century), the mathematics is so old that the original discoverers (or inventors?) are unknown. This kind of work is typical of the algebra you study in pre-college mathematics and the field of algebra itself up to about the start of the 19th century. At that time the ``state of the art'' was roughly as follows:

It was known that quadratic, cubic, and quartic polynomial equations could be solved in general by ``radical'' formulas like the quadratic formula, but
On the other hand, (by results of Abel, Ruffini, and Galois that was proved by means of ideas we will see later this semester) it was also known that the ``general'' polynomial equation of degree 5 or greater is not solvable in this fashion.

The work of Abel, Ruffini, and Galois, as well as subsequent work of Cayley, Sylvester, Hamilton, Boole, etc. was the start of what is now known as ``modern'' or ``abstract'' algebra. The basic idea of modern algebra is not only to study how to manipulate expressions to solve for a variable, but also to focus on the underlying algebraic structures that allow you to make those manipulations. This is what made it possible to prove statements like the unsolvability of the general polynomial of degree at least 5 by radicals.

To understand what this abstract approach means, consider the following. In solving quadratic equations by the quadratic formula, we usually assume that the coefficients a,b,c are real numbers. Then the ``usual rules of algebra'' for the addition and multiplication of real numbers - the commutativity and associativity of addition and multiplication, the distributive law for multiplication over addition, the existence of multiplicative inverses for nonzero a, the existence of square roots for all non-negative real numbers, etc. - are what allow us to derive the quadratic formula. Listing all these properties of addition of real numbers provides the definition of the algebraic structure known as a group. We also have the first glimmer of the way the facts about solutions of equations metioned above were developed if we note that interchanging +/- in the quadratic formula leaves the set of roots of a quadratic equation unchanged(!)

But in fact much more insight is gained if we study groups ``in the abstract'' and ask:

what examples of groups are there?
what properties do they all have?
what groups act on the sets of roots of equations?

The topics we will be studying are:

The basic language of group theory, first examples, subgroups, cosets, Lagrange's Theorem (about 11 days)
Normal subgroups, factor groups, homomorphisms, isomorphisms (about 8 days)
Structure theory for finite and finitely-generated abelian groups (about 5 days)
Symmetric and alternating groups, simple groups (about 4 days)
The Sylow theorems and applications (about 9 days)

Objectives

The major objectives of the course are:

To introduce the algebraic structures called groups, and to continue the study of mathematical structures begun in MATH 243,
To show how these structures arise from many of the basic problems of algebra and how they underlie many other topics in mathematics,
To introduce you to some of the history and development of algebra as a mathematical discipline,
To continue the development of your problem-solving skills in mathematics,
To continue the development of your skills in reading, developing, and writing mathematical proofs.

Text

The text for the course is Abstract Algebra, An Introductory Course, by Gregory T. Lee, Springer, ISBN 978-3-319-77648-4. All of the course readings and most of the problems this semester will come from this text. We will be studying the material in Chapters 2 - 7. (Chapter 1 reviews some topics that should be familiar from MATH 243. You may want to refer to that from time to time to jog your memory on definitions related to functions, injectivity, surjectivity, equivalence relations, etc.)

Course Assignments and Grading

The assignments for the course will consist of:

Two in-class midterm exams, each worth 20% of the course grade. Tentative dates: Friday, October 5 and Friday, November 16. I am happy to consider administering these in two-hour blocks on the preceeding Thursday evenings rather than in the 50-minute regular class period if the class would prefer to take them that way.
Final Examination, worth 30% of the course grade. Scheduled date will be announced by the Registrar after the start of the semester. Please do not make travel plans for the semester break until you know when our exam is scheduled.
Weekly problem sets, worth 20% of the course grade. Notes:
- Because of the relatively large size of this class, in order for me to return your work with constructive comments in a timely manner, it may become necessary to grade only selected problems on each assignment. If that happens, I will always pick a representative sample of problems to be evaluated from that assignment. But the selection will not be announced beforehand, and you will be expected to do all of the problems in any case.
- Because of every effort will be made to return your graded problem sets in a timely fashion, and for reasons of fairness, no problem sets will be accepted for credit after the announced due date, except in the case of a verified medical excuse. If you are authorized to hand in a problem set late, I will ask you sign a statement that you have not consulted the reserve solutions in preparing your work.
Oral problem presentations to the class worth 10% of the course grade. Each student will do two of these, on assigned problems, on the designated ``problem days.''

I will be keeping your course average in numerical form throughout the semester, and only converting to a letter for the final course grade. The course grade will be assigned according to the following conversion table (also see Note below):

A -- 94 and above
A- -- 90 - 93
B+ -- 87 - 89
B -- 84 - 86
B- -- 80 - 83
C+ -- 77 - 79
C -- 74 - 76
C- -- 70 - 73
D+ -- 67 -- 69
D -- 60 - 66
F -- 59 and below.

Note: Depending on how the class as a whole is doing, some downward adjustments of the above letter grade boundaries may be made. No upward adjustments will be made, however. (This means, for instance, that an 85 course average might end up corresponding to a letter grade of B+ if the overall class average is lower than the scale above. But an 85 would never convert to a letter grade of B- or below.)

If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.

Schedule

A detailed, day-by-day schedule will be maintained on the course homepage for you to consult as needed. You should always treat that as an approximation, though. Some rearrangement, addition, expansion, or contraction of topics may become necessary as the semester progresses. I will announce any changes in class, and on the course homepage.

Departmental Statement on Academic Integrity

Why is academic integrity important?

All education is a cooperative enterprise between teachers and students. This cooperation works well only when there is trust and mutual respect between everyone involved. One of our main aims as a department is to help students become knowledgeable and sophisticated learners, able to think and work both independently and in concert with their peers. Representing another person's work as your own in any form (plagiarism or ``cheating''), and providing or receiving unauthorized assistance on assignments (collusion) are lapses of academic integrity because they subvert the learning process and show a fundamental lack of respect for the educational enterprise.

How does this apply to our courses?

You will encounter a variety of types of assignments and examination formats in mathematics and computer science courses. For instance, many problem sets in mathematics classes and laboratory assignments in computer science courses are individual assignments. While some faculty members may allow or even encourage discussion among students during work on problem sets, it is the expectation that the solutions submitted by each student will be that student's own work, written up in that student's own words. When consultation with other students or sources other than the textbook occurs, students should identify their co-workers, and/or cite their sources as they would for other writing assignments. Some courses also make use of collaborative assignments; part of the evaluation in that case may be a rating of each individual's contribution to the group effort. Some advanced classes may use take-home examinations, in which case the ground rules will usually allow no collaboration or consultation. In many computer science classes, programming projects are strictly individual assignments; the ground rules do not allow any collaboration or consultation here either.

What are the responsibilities of faculty?

It is the responsibility of faculty in the department to lay out the guidelines to be followed for specific assignments in their classes as clearly and fully as possible, and to offer clarification and advice concerning those guidelines as needed as students work on those assignments. The Department of Mathematics and Computer Science upholds the College's policy on academic honesty. We advise all students taking mathematics or computer science courses to read the statement in the current College catalog carefully and to familiarize themselves with the procedures which may be applied when infractions are determined to have occurred.

What are the responsibilities of students?

A student's main responsibility is to follow the guidelines laid down by the instructor of the course. If there is some point about the expectations for an assignment that is not clear, the student is responsible for seeking clarification. If such clarification is not immediately available, students should err on the side of caution and follow the strictest possible interpretation of the guidelines they have been given. It is also a student's responsibility to protect his/her own work to prevent unauthorized use of exam papers, problem solutions, computer accounts and files, scratch paper, and any other materials used in carrying out an assignment. We expect students to have the integrity to say ``no'' to requests for assistance from other students when offering that assistance would violate the guidelines for an assignment.

Specific Guidelines for this Course

Because of the large size of this class, examinations will be given in scheduled, proctored sessions. No sharing of information in any form with other students will be permitted during exams. The other assignments will be the weekly individual problem sets and the oral presentations. On the problem sets, discussion of the questions with other students in the class, and with me during office hours is allowed, even encouraged. Your final problem write-ups should be prepared individually, however, and the wording and organization of the writeup should be entirely your own work. If you take advantage of any of the options described above for consultation on the problems, you will be required to state that fact in a "footnote" accompanying the problem solution. Failure to follow this rule will be treated as a violation of the College's Academic Integrity policy.