The College of the Holy Cross

Course Description

This semester, we will concentrate on the theory of groups, an area of abstract algebra that has connections with and applications to many other branches of mathematics, computer science, physics, etc. There will also be a strong connection with many of the ideas you encountered in linear algebra. Students who have taken the Algebraic Structures class will have seen the definition of a group and some of the basics. We will review that at the start for those who did not take Algebraic Structures. However, we will concentrate different kinds of examples of groups in this class, and go into group theory in much greater detail.

The topics we will be studying are:

1. The basic language of group theory, first examples, homomorphisms, normal subgroups, and quotient groups (about 10 class days)
2. Actions of groups on sets and the property of symmetry or invariance under a group action (about 6 days)
3. The ``fine structure'' of finite groups -- conjugacy classes and the Sylow theorems, algorithmic techniques for group theory (the Todd-Coxeter algorithm) (about 14 days)
4. Matrix, or linear groups -- one place where algebra and geometry come together (about 9 class days)

Textbook

The text for the course is Algebra by Michael Artin. (Catchy title!) As you will see, this book is actually designed for an integrated three-semester algebra sequence including a semester of Linear Algebra. We will not cover the L.A. chapters, but they (together with A Course in Linear Algebra) will form a handy review of those topics if you need it. We will cover most of the material in Chapters 2,5,6, and 8 this term, and continue on to finish the text next semester. One of the reasons I chose this book for our course is the way it consistently makes connections between algebra and other areas of mathematics. I think it will be good for you to see these links. Please feel free at any time to direct any comments, complaints, praise (?) etc. about the book to me.

Assignments and Grading

The only way to really learn advanced mathematics is to work out and present solutions to challenging problems, collaboratively or individually. Thus the focus of this course will be a series of problem sets, given out roughly every other week. In addition, there will be two larger summary problem sets, one at midterm, the other at the end of the semester. There will be no in-class exams, but over the course of the semester I will ask each student to present two problem solutions to the seminar. The problems will come from the problem sets, and I will assign the presentations when the problem set goes out. You will always have adequate lead time to prepare for these presentations.

Beside the problem sets, the other assignments for the course will be group write-ups from collaborative group discussions. For some of the class meetings, we will break the class down into groups of three or four students to work on group discussion questions. These will lead to group write-ups, and on some occasions to oral reports to the rest of the class at the end of the period.

Your final grade for the seminar will be computed using the following weight factors:

If you ever have a question about an assignment, or about your standing in the course, please feel free to consult me, either during regular office hours, or by making an appointment to see me.