### Abstract Algebra

When doing your homework, I encourage you to use the Elements of Style for Proofs (see Appendix B of the course notes) as a reference. **Note:** On each homework assignment, please write (i) your name, (ii) name of course, and (iii) Daily/Weekly Homework number.

## Daily Homework

The following assignments are to be turned in at the end of the indicated class period. I reserve the right to modify the assignment if the need arises. These exercises will form the basis of the student-led presentations each day. Daily assignments will be graded on a $\checkmark$-system. During class, **you are only allowed to annotate your homework using the marker pens that I provide**.

**Daily Homework 1:**Read the syllabus and write down 5 important items.*Note:*All of the test dates only count as one item. Turn in on your own paper at the beginning of class. (Due Wednesday, August 27)**Daily Homework 2:**Read Chapter 1: Introduction of the course notes. In addition, complete 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 2.9, 2.10, 2.11, 2.12 in Chapter 2: An Intuitive Approach to Groups and digest the surrounding text along the way. (Due Wednesday, August 27)**Daily Homework 3:**Stop by my office (AMB 176) and say hello. If I’m not there, just slide note under my door saying you stopped by. (Due Friday, August 29)**Daily Homework 4:**Read Appendix B: Elements of Style of Proof. Also, complete 2.13, 2.15-2.25 in Chapter 2: An Intuitive Approach to Groups and digest the surrounding text along the way. (Due Friday, August 29)**Daily Homework 5:**Read Appendix C: Fancy Mathematical Terms. Also, complete 3.1-3.10 in Chapter 3: Cayley Diagrams and digest the surrounding text along the way. (Due Wednesday, September 3)**Daily Homework 6:**Read Appendix D: Definitions in Mathematics. Also, complete 3.11-3.17 in Chapter 3: Cayley Diagrams and digest the surrounding text along the way. (Due Friday, September 5)**Daily Homework 7:**Complete 4.1-4.3, 4.6, 4.7, 4.8 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Monday, September 8)**Daily Homework 8:**Complete 4.10-4.14 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Wednesday, September 10)**Daily Homework 9:**Complete 4.15, 4.18 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Friday, September 12)**Daily Homework 10:**Complete 4.19-4.27 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Monday, September 15)**Daily Homework 11:**Complete 5.8-5.11, 5.13, 5.14, 5.16, 5.17 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Wednesday, September 17)**Daily Homework 12:**Complete 5.20-5.30 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. You should also read and digest Definition 5.31 and Theorem 5.32. (Due Friday, September 19)**Daily Homework 13:**Read Section 5.3 in Chapter 5: A Formal Approach to Groups and complete 5.40-5.45. You’ll notice that I did not assign several problems and theorems in Section 5.3, but you more or less did these on the take-home portion of Exam 1. Be sure to re-read them! (Due Friday, September 26)**Daily Homework 14:**Read Sections 5.4 and 5.5 in Chapter 5: A Formal Approach to Groups and complete 5.46-5.50, 5.52, 5.54, 5.55. (Due Monday, September 29)**Daily Homework 15:**Read Section 5.5 in Chapter 5: A Formal Approach to Groups and complete 5.56-5.60, 5.62-5.63. (Due Wednesday, October 1)**Daily Homework 16:**Read Section 5.5 in Chapter 5: A Formal Approach to Groups and complete 5.65-5.68. (Due Friday, October 3)**Daily Homework 17:**Read Sections 5.5 and 5.6 in Chapter 5: A Formal Approach to Groups and complete 5.69-5.70. And get caught up on any outstanding problems if you need to. (Due Monday, October 6)**Daily Homework 18:**Read Section 5.6 in Chapter 5: A Formal Approach to Groups and complete 5.71-5.74, 5.76-5.78. (Due Wednesday, October 8)**Daily Homework 19:**Read Section 5.6 in Chapter 5: A Formal Approach to Groups and complete 5.79, 5.80, 5.82, 5.83. (Due Friday, October 10)**Daily Homework 20:**Read Section 6.1 in Chapter 6: Families of Groups and complete 5.85, 6.1, 6.2, 6.4-6.10. (Due Monday, October 13)**Daily Homework 21:**Read Section 6.1 in Chapter 6: Families of Groups and complete 6.12-6.14. (Due Wednesday, October 15)**Daily Homework 22:**Read Section 6.1 in Chapter 6: Families of Groups and complete 6.15-6.17. (Due Friday, October 17)**Daily Homework 23:**Read Section 6.1 in Chapter 6: Families of Groups and complete 6.27-6.29. (Due Friday, October 24)**Daily Homework 24:**Read Section 6.1 in Chapter 6: Families of Groups and complete 6.40-6.44, 6.47. Also, digest the meaning of Theorem 6.39, which you will need to do a few of the problems. I suggest you crank out a few examples to convince yourself the theorem is true. I’ll either prove this theorem in class next week or send out a proof for you to read at your leisure. (Due Monday, October 27)**Daily Homework 25:**Read Section 6.2 in Chapter 6: Families of Groups and complete 6.49-6.53. (Due Wednesday, October 29)**Daily Homework 26:**Read Section 6.3 in Chapter 6: Families of Groups and complete 6.55-6.56, 6.58-6.69. This looks like a lot, but most of them you should be able to dispense with rather quickly. (Due Friday, October 31)**Daily Homework 27:**Read Section 6.3 in Chapter 6: Families of Groups and complete 6.70-6.76, 6.79, 6.80. To do the last two problems, you’ll need to understand Theorem 6.78 and the discussion that follows it. (Due Monday, November 3)**Daily Homework 28:**Read Section 6.4 in Chapter 6: Families of Groups and complete 6.81-6.88, 6.90, 6.92-6.94, 6.97, 6.98. Also, make sure you read and digest Theorem 6.96. (Due Wednesday, November 5)**Daily Homework 29:**If necessary, finish up any problems from Daily Homework 28. Also, read Section 7.1 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and complete 7.2, 7.3, 7.4(a)(b), 7.5, 7.7, 7.8(2). You are also responsible for digesting the content of the problems that were omitted, but you do not have to formally complete them. (Due Friday, November 7)**Daily Homework 30:**Re-read Section 7.1 and read Section 7.2 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and complete 7.12, 7.13-7.19, 7.21, 7.22. You are also responsible for digesting the content of the problems that were omitted, but you do not have to formally complete them. (Due Monday, November 10)**Daily Homework 31:**Read Section 7.3 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and complete 7.24-7.31. (Due Wednesday, November 12)**Daily Homework 32:**Read Sections 8.1 and 8.2 in Chapter 8: Products and Quotients of Groups and complete 8.1, 8.6, 8.7, 8.26-8.30, 8.32, any 2 parts of 8.33. You are also responsible for digesting the content of the problems that were omitted, but you do not have to formally complete them. (Due Friday, November 14)**Daily Homework 33:**Complete the exercises found here (but skip the last problem). (Due Monday, November 24)**Daily Homework 34:**Complete the exercises found here. (Due Wednesday, November 26)**Daily Homework 35:**Complete the exercises found here. (Due Monday, December 1)**Daily Homework 36:**Complete the exercises found here. (Due Wednesday, December 3)**Daily Homework 37:**Last one! Complete the exercises found here (Due Friday, December 5)

## Weekly Homework

In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit 2-3 formally written solutions/proofs. You are required to type your submission using $\LaTeX$ (see below for more on this). I will walk you through how to do this.

**Weekly Homework 1:**Prove Theorem A.8 and either Theorem A.74 or Theorem A.75 from Appendix A. You are required to type your proofs using LaTeX. For more information, see below. You should email me the PDF of your completed work. (Due Thursday, September 4 by 8PM)**Weekly Homework 2:**Prove Theorem A.12 and Theorem A.43 from Appendix A. For Theorem A.43, it may be helpful to read Definition A.40. You are required to type your proofs using LaTeX and you should email me the PDF of your completed work. (Due Tuesday, September 9 by 8PM)**Weekly Homework 3:**Complete each of the following tasks. (Due Tuesday, September 16 by 8PM)- Prove either of Theorem 4.6 or Theorem 4.8.
- Determine whether each of the following statements is true or false. If a statement is true, write a short proof. If a statement is false, justify your reasoning. In each case, the context should make it clear what each letter represents. In particular, in items 1 and 3, $r$ represents rotation of a square by a quarter turn clockwise. But in item 4, $r$ represents rotating a triangle by a third of a turn clockwise.
- $\{s, r, sr, rs\}\leq D_4$
- $\{1, -1, i, -i, j, -j\}\leq Q_8$
- $\{e, sr, rs, r^2\}\leq D_4$
- $\{e, r, r^2\} \leq D_3$

**Weekly Homework 4:**Prove one of Theorems 5.24, 5.25, 5.28 and prove one of Theorems 5.30, 5.32(2) from Chapter 5: A Formal Approach to Groups. (Due Tuesday, September 30 by 8PM)**Weekly Homework 5:**Prove any two of Theorems 5.52, 5.58, 5.60, 5.62, 5.64 from Chapter 5: A Formal Approach to Groups. (Due Tuesday, October 7 by 8PM)**Weekly Homework 6:**Prove any two of Problem 5.76, Theorems 5.79, 5.80, 5.82, 5.83 from Chapter 5: A Formal Approach to Groups. (Due Tuesday, October 14 by 8PM)**Weekly Homework 7:**Prove any two of 6.14, 6.18, 6.19, 6.20, 6.22, 6.52(4) from Chapter 6: Families of Groups. (Due Tuesday, November 4 by 8PM)**Weekly Homework 8:**Prove any two of 6.63, 6.69, 6.70, 6.86 from Chapter 6: Families of Groups. If you choose to prove 6.86, you should include your constructions from 6.84 and 6.85. (Due Thursday, November 13 by 8PM)**Weekly Homework 9:**Prove any two theorems from Daily Homework 35-36. This includes Theorems 1, 2, 3 from Daily Homework 33, Exercise 7 from Daily Homework 34, Theorems 1, 2, 3 from Daily Homework 35, and any theorem-type problems from Daily Homework 36(Due Thursday, December 4 by 8PM)

## Using LaTeX for Weekly Homework

You are required to use $\LaTeX$ to type up your Weekly Homework assignments. To do this, I suggest that you use my LaTeX Homework Template. The easiest way to get started with $\LaTeX$ is to use an online editor. I recommend using writeLaTeX, but there are other options. The good folks over at writeLaTeX have preloaded my template, so to get started, all you need to do is click the link below.