# Math 31: Topics in Algebra

Last updated August 10, 2012

# Homework

Homework will be assigned most weeks, for a total of 8 assignments. Assignments will generally be given on Fridays, due the following Friday at the beginning of class. Homework solutions should be written legibly in pen or pencil or typed. All proofs should be written in complete sentences. I reserve the right to reject any papers that are illegible. Also, please be sure to staple all papers before handing them in.

The homework assignments will consist of both proofs and computational problems. Many of the problems will come from the textbook, but there will also be many problems of my own. The assignments will be divided into three sections corresponding to the difficulty of the problems:

• Easy: These are fairly basic problems that are mainly intended to check you understanding of the key concepts.

• Medium: These are slightly more difficult problems that will require you to apply the concepts you have learned to more interesting situations. The problems in this range are of the type that you should expect to see on the exams.

• Hard: These are fairly difficult problems. They will either require you to explore certain concepts beyond what we have done in class, or their solution may involve some creativity.

In general, an attempt at writing a mathematical proof either ends with a correct proof, or it does not. In other words, the solution to a proof question is either right or wrong. This idea will inform the grading of the homework problems. Each problem will be graded on a scale of 0 to 2, depending on whether the proof is correct, incorrect, or somewhere in between:

• 2: Your solution is a correct proof of the given statement.

• 1: The solution is mostly on the right track, but there are some small defects that keep it from being an airtight argument.

• 0: The proof is incorrect.

Since one of our goals is to truly develop proof-writing skills, I want you to have the opportunity to learn from your mistakes, and to really learn the material by eventually writing correct proofs. Therefore, you will have the chance to rewrite up to three problems from each homework assignment for full credit. You may choose any three problems on which you did not receive full credit, and you can turn in updated solutions to these problems within one week of receiving the assignment back. I will grade the rewrites according the usual guidelines, and if the rewrite merits a higher score, I will simply replace the original score on that problem with the new one.

Notes on collaboration: As noted in the honor statement for the course, I encourage you to talk to your classmates when working on the homework assignments. When learning abstract mathematics, it is often quite helpful to discuss ideas with others, and it can be easier to discern what one does and does not understand when trying to explain things to others. Therefore, collaboration is a valuable learning tool. However, you should keep the following guidelines in mind when working on homework.

• The solutions that you turn in to me should be written up by you in your own words. It is fine to discuss ideas with others, but I want each person to think individually about how to put those ideas down on paper.

• If you have worked with others on a particular problem, say so when you write up the solution. If you got a particular idea from someone else, give them the appropriate credit.

In summary, I encourage you to work on homework together, and to possibly work out ideas in groups. However, I do not want you to write up complete proofs as a group - this should be done individually. Look here for examples of what I think are acceptable levels of collaboration.

### Homework announcements

• The first homework assignment is now posted: Homework 1.pdf. It is due on Friday, June 29.
• The second homework assignment is now posted: Homework 2.pdf. It is due on Friday, July 6. (Update: There were two minor typos on the original version of the homework. They have been corrected.)
• The third homework is now out: Homework 3.pdf. It covers cyclic groups and subgroups, and cosets and Lagrange's theorem. For those of you who type your homework, here is the Latex code: Homework 3.tex.
• The fourth homework is now posted: Homework 4.pdf. It mostly deals with homomorphisms, and is due on Friday, July 20th. Here is the Latex code: Homework 4.tex
• The fifth homework is now posted: Homework 5.pdf. It mostly deals with homomorphisms, and is due on Friday, August 3rd. Here is the Latex code: Homework 5.tex
• The sixth homework is now posted: Homework 6.pdf. It is due on Friday, August 10th. Here is the Latex code: Homework 6.tex
• The seventh (and final!) homework is now posted: Homework 7.pdf. It is due on Friday, August 17th, and should be a little shorter than the previous assignments. Here is the Latex code: Homework 7.tex.

# Homework Schedule

Date
Assignment(s) due
Notes
6/22
6/23
6/25
6/27
6/29 Homework 1 is due today.
7/2
7/6 Homework 2 is due today.
7/9 You should be ready to tell me your groups for the project by today.
7/11
7/13 Homework 3 is due today. Here is the Latex code: Homework 3.tex
7/16
7/18
7/20 Homework 4 is due today. Latex code: Homework 4.tex
7/23
7/25
7/27
7/30
8/1 Rough outlines for the projects are due.
8/3 Homework 5 is due today. Latex code: Homework 5.tex
8/6
8/8
8/10 Homework 6 is due today. Latex code: Homework 6.tex
8/13 Detailed project outline is due.
8/15
8/17 Homework 7 is due. Latex code: Homework 7.tex
8/20
8/22 Individual reports for the projects are due, as well as rewrites for Homework 6.