# Math 375:Abstract Algebra I

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

Fall, 2005.

Instructor:  David J. Eck.

Monday, Wednesday, Friday:  11:15 AM -- 12:10 PM.
Room Napier 201
```

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 almost the entire 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 Abstract Algebra: A First Course, by Dan Saracino. The first 150 pages of this book cover group theory, while the rest of the book is an introduction to rings and fields, which are additional areas of abstract algebra. We will cover the group theory part of the book, although at the end of the course we might substitute a short introduction to rings and fields for some of the more advanced material on group theory.

### Homework Assignments

Homework will be assigned weekly. Most of the homework exercises will be taken from the textbook, but I will add a few problems of my own from time to time.

Your solution to every homework exercise must include a justification for your answer. This is obvious for exercises that ask you to "show" or "prove" something, but it is true for every exercise -- even for questions that have a yes/no answer. You will find the answers to many of the textbook's exercises at the back of the book. This will not necessarily stop me from assigning these exercises, since what I expect from you is a justification, not a simple one-word answer.

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.

In addition to the homework that I assign to the entire class, I might occasionally assign a problem to an individual, for that individual to work on and present in class.

There will be three in-class tests and a final exam. The tests will be given on Monday, September 26; Wednesday, October 26; and Wednesday, November 30. The final exam will be during the scheduled final examination period for this course, 1:30 PM on Thursday, December 15. 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.

```            First Test:     15%
Second Test:    15%
Third Test:     15%
Final Exam:     20%
Assignments:    35%
```

### 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/. I will post weekly readings and assignments on that page.

### Tentative Schedule

Aug. 29, 31; Sep. 2 Chapters 0 and 1; Begin Chapter 2
Sep. 5, 7, 9 Finish Chapter 2; Chapter 3
Sep. 12, 14, 16 Chapter 4
Sep. 19, 21, 23 Chapter 5
Sep. 26, 28, 30 Chapter 6
Test Monday, Chapters 0 though start of 5
Oct. 3, 5, 7 Chapter 7; Begin Chapter 8
Oct. 12, 14 Finish Chapter 8;
No class Monday due to Fall Break
Oct. 17, 19, 21 Chapter 9
Oct. 24, 26, 28 Begin Chapter 10
Test Wednesday, Chapters 5 through 9
Oct. 31; Nov. 2, 4 Finish Chapter 10, Begin Chapter 11
Nov. 7, 9, 11 Finish Chapter 11; Begin Chapter 12
Nov. 14, 16, 18 Finish Chapter 12; Begin Chapter 13
Nov. 21 Finish Chapter 13
No class Wednesday or Friday due to Thanksgiving
Nov. 28, 30; Dec. 2 Chapter 14
Test Wednesday, Chapters 10 through 13
Dec. 5, 7, 9 Chapter 15 or Selected Topics
Dec. 15 Final Exam
Thursday, December 15, 1:30 PM