On each homework assignment, please write (i) your name, (ii) name of course, and (iii) homework number. You are allowed and encouraged to work together on homework. Yet, each student is expected to turn in his or her own work. In general, late homework will not be accepted. However, you are allowed to turn in three late homework assignments with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late. 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.

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 colored marker pens that I provide**.

**Daily Homework 1:**Read the syllabus and write down 5 important items.*Note:*All of the exam dates only count as a single item. Turn in on your own paper at the beginning of class. (Due Friday, January 22)**Daily Homework 2:**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, January 22)**Daily Homework 3:**Read and sign the Student Contract. (Due Friday, January 22)**Daily Homework 4:**Read Chapter 1: Introduction of the course notes. In addition, complete 2.2-2.6, 2.8-2.12 in Chapter 2: An Intuitive Approach to Groups and digest the surrounding text along the way. (Due Friday, January 22)**Daily Homework 5:**Skim through Appendix B: Elements of Style of Proof of the course notes. In addition, complete 2.13, 2.15-2.25 in Chapter 2: An Intuitive Approach to Groups and digest the surrounding text along the way. (Due Monday, January 25)**Daily Homework 6:**Skim through Appendix C: Fancy Mathematical Terms of the course notes. In addition, complete 3.1-3.10 in Chapter 3: Cayley Diagrams and digest the surrounding text along the way. (Due Wednesday, January 27)**Daily Homework 7:**Skim through Appendix D: Definitions in Mathematics of the course notes. In addition, complete 3.11-3.12 in Chapter 3: Cayley Diagrams and digest the surrounding text along the way. (Due Friday, January 29)**Daily Homework 8:**Complete 3.13-3.17 in Chapter 3: Cayley Diagrams and digest the surrounding text along the way. (Due Monday, February 1)**Daily Homework 9:**Complete 4.1-4.3, 4.5-4.8, 4.10-4.15 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. Recall that when you encounter a theorem, your job is to prove it. (Due Wednesday, February 3)**Daily Homework 10:**Complete 4.16, 4.19-4.28 in Chapter 4: An Introduction to Subgroups and Isomorphisms and digest the surrounding text along the way. (Due Friday, February 5)**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 Monday, February 8)**Daily Homework 12:**Complete 5.20-5.27 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Wednesday, February 10)**Daily Homework 13:**Complete 5.28-5.30 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Friday, February 12)**Daily Homework 14:**Complete 5.32-5.41 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Monday, February 15)**Daily Homework 15:**Complete 5.56(d), 5.61, 5.63, 5.64, 5.66, 5.67, 5.68 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Friday, February 26)**Daily Homework 16:**Complete 5.69-5.77 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Monday, February 29)**Daily Homework 17:**Complete 5.78, 5.80-5.88 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Wednesday, March 2)**Daily Homework 18:**Complete 5.89-5.91 in Chapter 5: A Formal Approach to Groups and digest the surrounding text along the way. (Due Friday, March 4)**Daily Homework 19:**Complete 6.1, 6.3-6.11 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Monday, March 7)**Daily Homework 20:**Complete 6.13-6.20 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Wednesday, March 9)**Daily Homework 21:**Complete 6.21-6.24, 6.26-6.31 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Friday, March 11)**Daily Homework 22:**Complete 6.37-6.48 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Wednesday, March 23)**Daily Homework 23:**Complete 6.50-6.54 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Friday, March 25)**Daily Homework 24:**Complete 6.56, 6.57, 6.59-6.66 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Monday, March 28)**Daily Homework 25:**Complete 6.67, 6.80-6.88 in Chapter 6: Families of Groups and digest the surrounding text along the way. For 6.67, you only need to do the ones that I didn’t do in class. (Due Wednesday, April 6)**Daily Homework 26:**Complete 6.90-6.92, 6.94-6.96, 6.98, 6.100-6.105 in Chapter 6: Families of Groups and digest the surrounding text along the way. (Due Friday, April 8)**Daily Homework 27:**Complete 7.2-7.8, 7.15 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and along the way read through 7.9-7.14 (these should look familiar from the take-home portion of Exam 2). (Due Monday, April 11)**Daily Homework 28:**I’ve fixed part (d) of Theorem 7.8, which I’d like you to try again. In addition, complete 7.18, 7.19, 7.21, 7.22 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and digest the surrounding text along the way. Also, be sure to look at 7.17 (which should look familiar from the take-home portion of Exam 2). (Due Wednesday, April 13)**Daily Homework 29:**Complete 7.24-7.31 in Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups and digest the surrounding text along the way. In addition, skim the rest of Chapter 7. (Due Friday, April 15)**Daily Homework 30:**Complete 8.6-8.8, 8.10, 8.11, 8.13, 8.15-8.18 in Chapter 8: Products and Quotients of Groups and digest the surrounding text along the way. In addition, read and digest 8.1-8.5, 8.9, 8.12, and 8.14. (Due Monday, April 18)**Daily Homework 31:**Complete 8.19-8.22, 8.25 in Chapter 8: Products and Quotients of Groups and digest the surrounding text along the way. (Due Wednesday, April 20)**Daily Homework 32:**Complete 9.9, 9.10, 9.14-9.18, 9.21-9.23 in Chapter 9: Homomorphisms and the Isomorphism Theorems and digest the surrounding text along the way. (Due Friday, April 29)**Daily Homework 33:**Complete 10.6-10.11, 10.15-10.18, 10.20, 10.25, 10.27 in Chapter 10: An Introduction to Rings and digest the surrounding text along the way. There are a few theorems that I did not assign you to prove. You should make sure you understand these and you are welcome to use them for later exercises. (Due Monday, May 2)**Daily Homework 34:**Complete 10.31, 10.37-10.39, 10.42, 10.44, 10.48, 10.49 in Chapter 10: An Introduction to Rings and digest the surrounding text along the way. There are a few theorems that I did not assign you to prove. You should make sure you understand these and you are welcome to use them for later exercises. (Due Wednesday, May 4)

In addition to the Daily Homework, we will also have Weekly Homework assignments. For most of these assignments, you will be required to submit 2-4 formally written proofs. Most of the time, two of the problems will come directly from the Daily Homework from the previous week. Any additional problems will likely be new. You will be required to type your submission using $\LaTeX$ (see below for more on this). You should email me the PDF of your completed work or turn in a hardcopy. If you email me the PDF, please name your file as `WeeklyX-Lastname.pdf`

, where `X`

is the number of the assignment and `Lastname`

is your last name. Notice there are no spaces in the filename.

**Weekly Homework 1:**Prove**two**of Theorem A.43, Theorem A.44, Theorem A.75, or Theorem A.81 from Appendix A. In addition, write up a solution to**one**of Exercise 2.13 or Exercise 2.19.*Note:*For the problems coming from Appendix A, you are welcome to consult external resources. (Due Tuesday, February 2 by 8PM)**Weekly Homework 2:**Complete each of the following tasks. You are required to type your proofs using LaTeX. You should email me the PDF of your completed work or turn in a hardcopy. (Due Tuesday, February 9 by 8PM)- Prove any
**two**of Theorems 4.6, 4.8, or 4.20. - 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, 3, and 5, $r$ 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.
- $\{s, r, sr, rs\}\leq D_4$
- $\{1, -1, i, -i, j, -j\}\leq Q_8$
- $\{e, sr, rs, r^2\}\leq D_4$
- $\{e, r, r^2\} \leq D_3$
- $\{e, r, r^2\} \leq D_4$

- Suppose $G$ is a finite group such that $G=\langle g_1,\ldots, g_n\rangle$. Consider the Cayley diagram for $G$ using $\{g_1,\ldots,g_n\}$ as a generating set. Prove that for each $i$, if we follow a sequence of (forward) arrows corresponding to $g_i$ out of $e$, we eventually end up back at $e$.

- Prove any
**Weekly Homework 3:**Prove**two**of Theorems 5.24, 5.25, 5.28. (Due Tuesday, February 16 by 8PM)**Weekly Homework 4:**Prove**two**of Theorems 5.63, 5.83, 5.84, 5.86, 5.87, 5.89, 5.90. In addition, prove that if $\phi:G_1\to G_2$ is a function between two groups that satisfies the homomorphic property (which may or may not be 1-1 or onto), then the set $K=\{g\in G_1\mid \phi(g)=e_2\}$ (where $e_2$ is the identity of $G_2$) is a subgroup of $G_1$. (Due Tuesday, March 8 by 8PM)**Weekly Homework 5:**Prove**two**of Theorems 6.7, 6.8(a), 6.8(b), 6.20, 6.21, 6.23. In addition, prove that if $\phi:G_1\to G_2$ and $K$ are as in Weekly Homework 4, then $\phi$ is 1-1 iff $K=\{e_1\}$ (where $e_1$ is the identity of $G_1$). (Due Tuesday, March 22 by 8PM)**Weekly Homework 6:**Prove**two**of Theorems 6.64, 6.71, 6.87 (both parts), 6.92. (Due Tuesday, April 12 by 8PM)**Weekly Homework 7:**Prove**two**of Theorems 6.100, 7.18, 7.29, 7.31. (Due Tuesday, April 19 by 8PM)

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 Overleaf, but there are other options. The good folks over at Overleaf have preloaded my template, so to get started, all you need to do is click the link below.

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MAT 232: Mathematical Reasoning

MAT 411: Abstract Algebra

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