The course described on this page ended December 15, 2005

## Math 375:

Abstract Algebra I

Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall, 2005. Instructor: David J. Eck (eck@hws.edu) Monday, Wednesday, Friday: 11:15 AM -- 12:10 PM. Room Napier 201 Course Handout: http://math.hws.edu/eck/courses/math375_f05.html Office Hours for Fall 2005: Monday 2:00 -- 3:00 Tuesday 1:00 -- 3:00 Wednesday 1:00 -- 3:00 Friday 10:00 -- 11:00

Assignments and other information for Math 375

will be posted on this page as the course is taught

during the Fall term of 2005.

## Final Exam: December 15

The final exam for this course is at 1:30 PM on Thursday, December 15, in our regular classroom. An information sheet is available. Students should also refer to the review sheets for Test 1, Test 2, and Test 3.

## Fifteenth Week: December 5, 7, and 9

In the final week of classes, we will finish a few topics from Chapter 13, and then move on to Chapter 14. From Chapter 14, you are responsible for the statement of the Fundamental Theorem of Finite Abelian Groups, but we will only cover part of the proof of this theorem. After finishing this material, I will spend whatever time is left over talking about some of the other aspects and applications of Abstract Algebra.

The homework from Chapter 13 is due on Wednesday.

## Fourteenth Week: November 28 and 30; December 2

There is a test on Friday, December 2; a review sheet is available.

Don't forget that the homework from Chapter 12, assigned two weeks ago, is due on Wednesday.

We will finish up Chapter 13 this week, even though not all of it is covered on the test. The following exercises from that chapter will be due next Wednesday, December 7. This will be the final graded homework assignment:

Chapter 13: # 13.4, 13.9, 13.11, 13.12, 13.13, 13.26

## Thirteenth Week: November 21

On Monday, we will continue with Chapter 12. There is no class on Wednesday or Friday this week, because of the Thanksgiving holiday. Happy Thanksgiving!

## Twelfth Week: November 14, 16, and 18

This week, we will be working on Chapter 12. The following homework will be due in two weeks, on Wednesday, November 30:

Chapter 12: # 12.2, 12.5, 12.7, 12.16, 12.20, 12.21, 12.22, 12.30, 12.34The third test was originally scheduled for Wednesday, November 30, but we have decided to postpone it for one class, it to Friday, December 2.

## Eleventh Week: November 7, 9, and 11

We have added Exercise 10.25 onto the homework that is due this Wednesday, November 9.

The reading for the week is Chapter 11, which covers normal subgroups and quotient groups. The following homework is due next Wednesday, November 16:

Chapter 11: # 11.2, 11.3, 11.5, 11.7, 11.8, 11.12, 11.15, 11.16, 11.18

## Tenth Week: October 31; November 2 and 4

You should finish reading Chapter 9 and read Chapter 10. The material in Chapters 9 through 13 is all closely related and is probably the most important part of the course. Chapter 9 defines cosets and Chapter 10 uses them to prove Lagrange's Theorem, that the order of a subgroup divides the order of the group. This important theorem has several interesting consequences. The following homework is due next Wednesday, November 9:

Chapter 9: # 9.5, 9.6, 9.8, 9.11 Chapter 10: # 10.2, 10.5, 10.6, 10.7, 10.8, 10.9

## Ninth Week: October 24, 26, and 28

There is a test on Wednesday of this week. It will cover Chapters 5 through 8. A review sheet is available.

We did not even begin Chapter 9 last week and, in fact, we still have some material to cover from Chapter 8. On Monday, we will finish up Chapter 8 and review for the test. We will begin Chapter 9 on Friday. There is no homework assignment due next week.

## Eighth Week: October 17, 19, and 21

The reading for the week is Chapter 8 and pages 79 to 81 in Chapter 9. The pages in Chapter 9 cover equivalence relations only. There is a test coming up next week. Because of the test, there will be no homework assignment due next week. However, it would be good to do Exercises 9.1, 9.2, and 9.3 for practice.

## Seventh Week: October 12 and 14

There is no class on Monday, because of Fall break. On Wednesday, we will finish up Chapters 6 and 7, and we will begin Chapter 8 on Wednesday or Friday. You can begin reading Chapter 8 and continue it next week. The following homework is due on

Fridayof next week, October 21:Chapter 8: # 8.1, 8.2, 8.4, 8.7, 8.8, 8.13, 8.14, 8.16, and 8.17

## Sixth Week: October 3, 5, and 7

The reading for the week is Chapters 6 and 7. Both of these chapters are short, and Chapter 7 covers background material that you should already be familiar with. The following homework problems are due next Friday, October 14:

Chapter 6: # 6.1, 6.3, 6.5, 6.6, 6.7, 6.9, and 6.11 Chapter 7: # 7.6, 7.7, 7.8

Note added late:Although I assigned problem 6.11 for homework, I actually had wanted to assign problem 6.14 instead. Unfortunately, I don't like problem 6.11 very much. So, you can do 6.14 instead of 6.11, if you like, but if you want to do 6.11, that will also be OK.(The due date for the homework is Firday, not Wednesday, because of Fall Break.)

## Fifth Week: September 26, 28, and 30

There is a test on Monday, September 26. A review sheet is available.

After the test, we will continue with Chapter 5. Homework exercises for this chapter will be posted here later in the week.

## Fourth Week: September 19, 21, and 23

There is a

testcoming up next Monday, September 26. It will cover everything from the textbook up to page 46. This includes Chapters 0 through 4 plus the first few pages of Chapter 5. In addition to the material from the book, you are responsible for the material that we covered on symmetries and the groups D_{n}.Because of the test, no homework will be collected next week. However, here are some suggested exercises from Chapter 5 that you will want to work on before the test. We might do some of these exercises in class on Friday:

Chapter 5, page 52: 5.1, 5.2, 5.3, 5.8, 5.10, 5.11

Don't forget the picnic on Tuesday at 5:00 PM!

## Third Week: September 12, 14, and 15

The reading for the week is Chapter 4, which is really the beginning of a serious investigation of groups. The following homework from this chapter is due next Wednesday, September 21:

Chapter 4, page 41: 4.5, 4.6, 4.11, 4.13, 4.15, 4.20, 4.21, 4.22, 4.25You are invited to the annual Fall Picnic of the Department of Mathematics and Computer Science, which will take place on Tuesday, September 20, at 5:00 PM in the Science Mini-quad.

## Second Week: September 5. 7, and 9

We finish Chapter 2 and cover all of Chapter 3 this week. These chapters cover the most basic properties of groups. We have also been talking about groups of symmetries, and there is a homework problem on this topic.

Here is the homework for the week, which is due next Wednesday, September 14:

Chapter 3, page 31: 3.3, 3.4, 3.5, 3.7, 3.11, 3.12, 3.14 Additional Exercises: (A) What is the point of Exercises 3.3 above? (Why did the textbook author think that this is an interesting example?) (B) Consider an infinitely long row of identical T's: . . . T T T T T T T . . . Describe the group of symmetries of this row of T's, considered as a subset of the plane. You might not come up with a complete specification of the group, but you should try to describe its elements and say what you can about the composition of some typical pairs of elements. (C) Repeat exercise (B) for an infinitely long row of identical H's: . . . H H H H H H H . . .

## First Week: August 29 and 31; September 2

Chapter 0 of the text is background information that you should already be familiar with, and Chapter 1 is also likely to be familiar. We will review the material in these two chapters during the first two days of the course. On Friday, we will start Chapter 2, which begins our study of Group Theory, the topic that will concern us for almost the entire course.

I will generally collect homework weekly. Here is the homework for the first week of the term:

Chapter 1, page 14: 1.2, 1.7, 1.8, 1.9 Chpater 2, page 22: 2.4b, 2.6, 2.7, 2.8, 2.12, 2.13This homework is due at the beginning of class, next Wednesday, September 7. I will hand out my own solutions to the homework at that time, so you must turn in the homework on time to get credit. You are permitted and encouraged to work with other people in the course on the homework, but you should write up your own solutions in your own words to turn in.

Please remember that you are expected to justify your answer in

allcases.

Derrick Moore(Hobart '05) will be the department's math intern this year. The math intern holds regular office hours in Lansing 310. Derrick took Math 375 and will be able to offer help to students in the course this term. His office hours (subject to change) are as follows:Monday through Thursday: 2:00 to 5:00 PM and 7:00 to 10:00 PM Friday: 1:00 to 4:00 PM Sunday: 6:30 to 10:30 PMI will know my own schedule of office hours by Wednesday of this week. For the first two days of the term, August 29 and 30, I will be in my office at least between 1:00 and 3:00 PM.