# Math 375:Abstract Algebra I

```   Department of Mathematics and Computer Science
Hobart and William Smith Colleges

Fall, 2004.

Instructor:  David J. Eck.

Monday, Wednesday, Friday, 1:55--2:50 PM.
Room Napier 101.
```

Abstract algebra is one of the core subjects of theoretical mathematics, and it is often the first place where students encounter fully abstract mathematical thinking. That is, the objects that are studied in abstract algebra are defined in a purely abstract way, by a set of axioms, and the subject consists of the exploration of the properties implied by those axioms. The tools are definitions, theorems, and proofs. Nevertheless, abstract algebra is inspired by the real world, and it has turned out to be one of the most applicable areas of theoretical math.

This is a course in group theory. We will spend the semester studying the mathematical entities known as groups. A group consists of a set together with a binary operation on that set. The operation must satisfy certain properties. Examples of groups include the integers with the operation of addition and the set of invertible 3-by-3 matrices of real numbers with the operation of matrix multiplication. Many groups can be defined in terms of symmetry of geometrical objects; in fact, group theory has been referred to as the mathematical study of symmetry.

The textbook for this course is Contemporary Abstract Algebra, Fifth Edition, by Joseph A. Gallian. This book has more than enough material for two courses in abstract algebra. We will spend the first twelve weeks of the course covering Parts 1 and 2 of the book (Chapters 0 through 11). These chapters cover the fundamentals of group theory. Parts 3 and 4 are about rings and fields, which would be covered in Abstract Algebra II; we will skip them entirely. Part 5 of the book is devoted to "special topics," including several topics in group theory. We will select several of these group theory topics to be covered during the last two weeks of the course.

### Homework Assignments

There will be weekly assignments, which will be due the following week. Most or all of the homework exercises will be taken from the textbook. I will assign even-numbered exercises, since answers to odd-numbered exercises are given at the end of the book. However, you will probably want to do some of the odd-numbered exercises for practice.

I encourage people to work together on homework. However, everyone is responsible for writing up and turning in their own solutions, and for understanding what they turn in. Occasionally, to ensure that everyone understands the work that they turn in, I will set up meetings in my office where students will present some of their homework solutions to me and answer questions about them.

Since the class is so small, it might be a good idea to set up a time when the entire class can meet to work on problems. We will discuss this possibility in class.

### Tests and Grading

There will be three in-class tests and a final exam. The tests will be given on Friday, September 24; Monday, October 25; and Monday, November 22. The final exam will be during the scheduled final examination period for this course, 1:30 PM on Thursday, December 16. It is possible that I will include a take-home component in one or more of the tests. The final exam will cover material from the entire course, with some emphasis on the last two weeks of the course.

Your grade for the course will be computed as follows:

```            First Test:     15%
Second Test:    15%
Third Test:     15%
Final Exam:     25%
Assignments:    30%
```

### Attendance

I expect you to be present and on time for all classes. There might be extraordinary circumstances that force you to miss a few classes. You should discuss any such cases with me at the earliest possible time. In the absence of such extraordinary circumstances, you can expect any absence to lower your grade for the course.

### Office Hours, E-mail, and Web

My office is room 301 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. I will announce my official office hours as soon as I schedule them.

My e-mail address is eck@hws.edu. E-mail is good way to communicate with me, since I usually answer messages the day I receive them.

The Web page for this course is at http://math.hws.edu/eck/math375/f04. I will post weekly readings and assignments on that page.

### Tentative Schedule

Aug. 30; Sep. 1, 3 Chapter 0; Begin Chapter 1
Sep. 6, 8, 9 Finish Chapter 1; Chapter 2
Sep. 13, 15, 17 Chapter 3
Sep. 20, 22, 24 Chapter 4
Test on Friday on Chapters 0--4
Sep. 27, 29; Oct. 1 Chapter 5
Oct. 4, 6, 8 Chapter 6
Oct. 13, 15 Chapter 7;
No class Monday due to Fall Break
Oct. 18, 20, 22 Chapter 8
Oct. 25, 27, 29 Begin Chapter 9
Test Monday on Chapters 5--8
Nov. 1, 3, 5 Finish Chapter 9; Begin Chapter 10
Nov. 8, 10, 12 Finish Chapter 10
Nov. 15, 17, 19 Chapter 11
Nov. 22 Test Monday on Chapters 9--11;
No class Wednesday or Friday due to Thanksgiving
Nov. 29; Dec. 1, 3 Selected topics from Part 5 of the text
Dec. 6, 8, 10 Selected topics from Part 5 of the text
Dec. 16 Final Exam
Thursday, December 16, 1:30 PM