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.

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 colored marker pens that I provide.

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 2-4 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.
      1. $\{s, r, sr, rs\}\leq D_4$
      2. $\{1, -1, i, -i, j, -j\}\leq Q_8$
      3. $\{e, sr, rs, r^2\}\leq D_4$
      4. $\{e, r, r^2\} \leq D_3$
      5. $\{e, r, r^2\} \leq D_4$
    • 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 $e$.
  • 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)

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 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.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 232: Mathematical Reasoning
  MAT 411: Abstract Algebra

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