We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

Week 1

  • Monday, August 28: First day! I kicked off class by summarizing the first few homework assignments, which was followed by a whirlwind tour of the course webpage. After this, the students engaged in the first half of the Setting the Stage activity, which was followed by a short discussion of productive failure and productive struggle. The remaining portion of class was devoted to highlighting some key items on the syllabus.
  • Wednesday, August 30: The first few minutes of class were devoted to learning names and making sure there weren't any questions about the syllabus. We then split the class up into several small groups, where each group was responsible for one of problems from the homework. After a few minutes we came back together to discuss the problems. Our very first presentation was given by BW. She presented Exercise 2.2 and did a great job! With the time remaining we had AO, JC, and LC present Exercises 2.3, 2.5, and 2.9, respectively. Along the way, I discussed Exercise 2.4. I'm really pleased with how things went today. Unfortunately, we didn't get to Exercises 2.6, 2.8, 2.10-2.14. The plan is tackle these problems during the next class.
  • Friday, September 1: Another busy but productive day. I spent the first few minutes discussing few key points. This discussion included a summary of the rules for the intuitive definition of a group. The rest of class was devoted to covering ground in the notes. Along the way, MG, JG, DW, and AB2 presented Exercises 2.8, 2.10, 2.11, and 2.17, respectively. In addition, I lead discussions about Exercises 2.6, 2.12, 2.14, 2.18, and 2.19. Exercises 2.20-2.23 are still outstanding.

Week 2

  • Monday, September 4: Labor Day, no class!
  • Wednesday, September 6: I really enjoyed today! The first few minutes were devoted to discussing Exercises 2.20-2.23, which were left over from last week. This discussion included a review of the Rules 1-4 for the intuitive definition of a group. Next, we split the class up into 8 small groups, where each group was tasked with discussing one of Exercises 3.1, 3.2/3.3, 3.4, 3.5/3.6. We had SS, RF, NF, and DF present 3.1, 3.2/3.3, 3.4, and 3.5/3.6, respectively. We will squeeze in a quick discussion of Exercise 3.7 on Friday.
  • Friday, September 8: We cranked through a lot of stuff today. We kicked off with a general discussion of Cayley diagrams and then we chatted about Exercises 3.7 and 3.8. After small groups discussed problems, we had CK, GC, DV, and AO present Exercises 3.9(c), 3.10(cd), 3.11(bcd), and 3.11(fg), respectively. All the presentations were great. With the 10 minutes we had remaining, we spent some time discussing LaTeX, which students will use to type up their first weekly homework assignment.

Week 3

  • Monday, September 12: After a few quick announcements, we jumped right into student presentations. JN, EM, BD, and AB2 presented Exercises 3.12, 3.13, 3.14, and 3.15, respectively. All of the presentations generated good discussion. Next, I summarized the key points of Exercises 3.16 and 3.17. With the few minutes we had left, I discussed the use of LaTeX for the Weekly Homework.
  • Wednesday, September 13: Dr. Falk covered for me today while I was out. My understanding is that DW, JG, TR, JC, GF, JN, and DV presented Exercise 4.2, Exercise 4.3, Exercise 4.5, Theorem 4.6, Exercise 4.7, Theorem 4.8, and Exercise 4.11, respectively. The plan is to wrap up 4.12 and 4.13 next time.
  • Friday, September 15: Dr. Falk covered for me again today. BD, LC, SS, and JR presented Exercises 4.12, 4.13, 4.14, and Exercise 4.15 Our goal on Monday will be to wrap up discussion of 4.16, 4.19-4.24 and tackle 4.25-4.28.

Week 4

  • Monday, September 18: Today was action packed. We kicked off with reviewing some key ideas and looking over all of the Cayley Diagrams we have encountered so far this semester. As we marched through the notes, we had MG, JG, AO, and NF presented Exercise 4.19, Theorem 4.20, Problem 4.22, and Problem 4.25, respectively. I also led discussion on Problem 4.21, Exercise 4.23, Problem 4.24, and Problem 4.26, respectively.
  • Wednesday, September 20: After a few announcements, we split the class up into 7 small groups, where each group was tasked with discussing one of the homework problems. After some time, we had NF, BH, GL, AB1, JQ, BH, and MB present Exercise 4.27, Problem 4.28, Exercise 5.8, Exercise 5.9, Exercise 5.10, Exercise 5.11, and Exercise 5.16, respectively. During the last few minutes, I led a discussion about the proof of Theorem 5.14.
  • Friday, September 22: We started with a discussion of Exercises 5.21 and 5.22. Next, we had students present one at a time. TR, GC, GF, RF, GL, BW, DF, and AB1 presented Exercise 5.20, Exercise 5.23(ab), Exercise 5.23(cd), Exercise 5.23(ef), Exercise 5.23(gh), Exercise 5.23(ij), Theorem 5.24, and Theorem 5.25, respectively. We will tackle Exercise 5.26 and Corollary 5.27 next week.

Week 5

  • Monday, September 25: We didn't finish everything today, but we made good progress. We have BH, MG, HH, DW, EM, DW, and JG present Exercise 5.26, Corollary 5.27, Theorem 5.28, Theorem 5.29, Theorem 5.30, Theorem 5.32(a), and Theorem 5.30(b), respectively. We will try to squeeze Exercise 5.34 and Theorem 5.35 next time.
  • Wednesday, September 27: We spent quite a bit of time discussing the upcoming exam. After that we had WC, JC, JR, BD/EM, and CK present Exercise 5.34($D_3$), Exercise 5.34($S_3$), Exercise 5.34($D_4$), Exercise 5.34($Q_8$), and Exercise 5.38, respectively. Along the way, I also guided us through Theorem 5.35 and Exercise 5.37.
  • Friday, September 29: The students took the in-class portion of Exam 1.

Week 6

  • Monday, October 2: While the students are working on the take-home portion of Exam 1, I will lecture over stuff in the notes. Today, I discussed Problem 5.39, Problem 5.40, Problem 5.42, Problem 5.43, and Problem 5.45. We'll pick up where we left off on Wednesday.
  • Wednesday, October 4: After handing back the in-class portion of Exam 1, we had a quick discussion of a few of the commonly missed problems. Next, I continued lecturing over Chapter 5. In particular, I gave a quick summary of Section 5.4 and then covered Definition 5.52, Theorem 5.53, Remark 5.54, Exercises 5.55-5.58, Theorem 5.59, Problem 5.60, Theorem 5.61, and Problem 5.62.
  • Friday, October 6: Today was our first day back to student presentations since last week. We kicked off with some quick discussion about centers of groups, and then we split the class up into 7 small groups. After a few minutes, we had JG, DV, AO, AB1, MB, JN, and AB2 present solutions to Theorem 5.63, Exercise 5.64(abcd), Exercise 5.64(efgh), Exercise 5.64(ijk), Exercise 5.67, and Exercise 5.69. With the few minutes we had left, I proved closure for Theorem 5.66. We will wrap up Theorem 5.66 next time and address Theorem 5.68.

Week 7

  • Monday, October 9: After handing back the take-home portion of Exam 1, I finished proving Theorem 6.6 and then tackled Theorem 6.8. Next, we had NF, WC, DW, JR, and MG present Exercises 5.70, 5.71, 5.72, 5.73, and 5.74, respectively. I hope to discuss Problem 5.75 next time. We may not formally discuss Problems 5.76 and 5.77.
  • Wednesday, October 11: I got the room early and put skeleton proofs on the board for Problem 5.80, Exercise 5.81(a), Exercise 5.81(b), Exercise 5.81(c), and Problem 5.82. We kicked off with a discussion of how the homomorphic property follows from our intuitive understanding of isomorphisms. Next, we had pairs work on filling in the blanks on the skeleton proofs that I put on the board. After a few minutes, we had DV, JQ, EM, JN, and AO present Problem 5.80, Exercise 5.81(a), Exercise 5.81(b), Exercise 5.81(c), and Problem 5.82, respectively. We didn't get to Theorems 5.83-5.85 today, so we will have to squeeze them in on Friday.
  • Friday, October 13: I was kind of amazed that we got through everything today, especially in light of the energy level in the room. I gave students to chat in pairs about the proofs that were due today and then solicited volunteers to present. We had JN, AB2, JR, NF, and RF present proofs for Theorems 5.83, 5.84, 5.85, 5.86, and 5.87, respectively. With the few minutes we had left at the end of class I summarized the proof of Theorem 5.88.

Week 8

  • Monday, October 16: We covered a tremendous amount of material today. I assigned 14 people (a couple of which didn't show up) to specific problems and then gave them a few minutes to collaborate with a buddy and write something on the board. I kicked off with proving that the subset in question was nonempty for Theorem 5.89. Then we had BH, CK, GC, LC, GF, TR, JC, DF, GC, GL, SS, BW, and BD present Theorem 5.89 (closure), Theorem 5.89 (inverses), Exercise 6.1(d), Exercise 6.1(f), Exercise 6.2(b), Exercise 6.2(c), Theorem 6.3, Exercise 6.5, Exercise 6.6(k), Exercise 6.6(b), Exercise 6.6(f), Exercise 6.6(a), and Exercise 6.6(d), respectively.
  • Wednesday, October 18: Great day! We had AO, BH, GF, HH, JQ, LC, GL, and DF present Exercise 6.7, Exercise 6.8(a), Exercise 6.8(b), Exercise 6.8(c), Theorem 6.12, Corolllary 6.13, Exercise 6.14, and Exercise 6.15, respectively. Along the way, I discussed Theorems 6.9-6.11. We had some time at the end to get started on the next batch of problems. We were able to discuss 6.16-6.19.
  • Friday, October 20: We had TR, CK, AB1, EM, JC, RF, BW, and JG presented Theorem 6.20, Exercise 6.21/6.22, Theorem 6.23, Theorem 6.24, Theorem 6.25, Theorem 6.27 ($n=1,2$), and Theorem 6.27 ($n\geq 3$), respectively.

Week 9

  • Monday, October 23: We started with a discussion of Theorem 6.30 and deferred a proof of Theorem 6.31 until later. Next, we had MB, SS, HH, AB2, BD, and MG present Exercise 6.32, Exercise 6.33Exercise 6.34, Theorem 6.35, Theorem 6.36 ($n=1,2$), and Theorem 6.36 ($n\geq 3$), respectively. We will tackle Theorem 6.37 at the beginning of class on Wednesday.
  • Wednesday, October 25: We split the class up into several small groups, each tasked with discussing one of the homework problems. We had DW, JR, JN, GF, BH, DV, and BD share solutions to Theorem 6.37, Theorem 6.38 (forward direction), Theorem 6.38 (reverse direction), Exercise 6.44, Corollary 6.45, Exercise 6.46, and Exercise 6.47, respectively. With the few minutes we had left at the end, I blitzed through a proof of Theorem 6.40. We never got to Corollary 6.39, Exercise 6.41, and Corollary 6.42.
  • Friday, October 28: The students took the in-class portion of Exam 2.

Week 10

  • Monday, October 30: I lectured over the remaining bit of Section 6.1 and started Section 6.2. We got up to part (a) of Theorem 6.56.
  • Wednesday, November 1: I continued lecturing over Chapter 6. We wrapped up Section 6.2 and started Section 6.3. We got up to Problem 6.67.
  • Friday, November 3: After handing back the in-class portion of Exam 2, I discussed a few of the problems from the exam. Next, we picked up where we left off in the notes. We covered Theorem 6.68 through Theorem 6.75.

Week 11

  • Monday, November 6: I was out sick today and class ran without me. According to a report from AO, here is what transpired in my absence. DW, GF, MG, JN, RF, CK, EM, TR, DV, and JR presented Theorem 6.75, Problem 6.76, Exercises 6.77, 6.78, 6.79, 6.80, 6.81, 6.84, 6.85, and 6.86, respectively. We'll save some time on Wednesday to cover the highlights from these.
  • Wednesday, November 8: After handing back the take-home portion of Exam 2, I discussed Problem 3(b) from the exam. Next, I reviewed Theorem 6.75, Exercise 6.80, Theorem 6.83 (Cayley's Theorem), and Exercise 6.84, which were covered on Monday. With the time we had left, we had GC and BH present Exercises 6.87 and 6.88.
  • Friday, November 10: Veteran's Day! No class.

Week 12

  • Monday, November 13: I spent the first few minutes discussing the rigid and non-rigid symmetries of the cube. This was related to Remark 6.93. Then we had JG, JC, AB2, DF, LC, JN, and BD present Problem 6.89, Problem 6.90, Theorem 6.96, Exercise 6.99, Exercise 6.100, Problem 6.101, and Problem 6.102, respectively. Along the way, I sketched proofs of Theorem 6.91 and Corollary 6.92.
  • Wednesday, November 15: After discussing the symmetry groups of the tetrahedron, dodecahedron, and the icosahedron, I walked us through Exercises 6.106 and 7.4. Next, we had AO, DV, MG, and DW present Exercises 6.108, 7.5, 7.6, and 7.7, respectively.
  • Friday, November 17: Disappointing attendance today, but c'est la vie. After summarizing a few of the key theorems, we jumped in and tried to prove some stuff. We had BD, NF, AB1, DW, RF, and TR present Theorem 7.9(a), Theorem 7.10(reflexive), Theorem 7.10(symmetric), Theorem 7.10(transitive), Theorem 7.13(injective), and Theorem 7.13(surjective), respectively. Along the way, I proved Corollary 7.12 and Corollary 7.14.

Week 13

  • Monday, November 20: We cranked through several problems today. We had BW, JC, CK, JG, LC, DF, NF, and JN present Exercise 7.17, Theorem 7.19, Corollary 7.20, Problem 7.21, Exercise 7.23, Exercise 7.24, Exercise 7.26, and Theorem 7.29, respectively. Along the way, I proved Theorem 7.16 (Lagrange's Theorem) and tackled Problems 7.30 and 7.32 and proved Theorem 7.31.
  • Wednesday, November 22: We had decent turn out considering it was the Wednesday before Thanksgiving. We covered quite a bit. We had RF, JC, BH, MG, MB, JG, AB2, and GC present Exercise 8.6, Exercise 8.7, Exercise 8.8, Corollary 8.11, Exercise 8.13, Theorem 8.14, Exercise 8.17, and Exercise 8.18, respectively. I also discussed Theorem 8.19 and Problem 8.20. Along the way, we also discussed a few of the results that I did not assign. We will tackle Exercise 8.24 and 8.25 next time.
  • Friday, November 26: No classes, Thanksgiving Holiday!

Week 14

  • Monday, November 27: Today was mostly me lecturing. We covered 8.24 through 8.32. Along the way DV presented Exercise 8.27. We will pick up where we left off on Wednesday.
  • Wednesday, November 29: We had a fairly productive day, wrapping up Chapter 8 and getting most of the way through Chapter 9. We had JG, EM, SS, AB2, DF, GF, AO, and JQ present Theorem 8.36, Exercise 9.3, Theorem 9.4, Theorem 9.6, Theorem 9.7, Exercise 9.9, Exercise 9.10, and Theorem 9.12, respectively. In addition, I presented Exercise 8.33, Theorem 8.35, Problem 8.37, Theorem 8.39, Problem 8.40, and Exercise 9.2.
  • Friday, December 1: Students took the in-class portion of Exam 3.

Week 15

  • Monday, December 4: We wrapped up Chapter 9 and I started lecturing over Chapter 10.
  • Wednesday, December 6: I continued lecturing over Chapter 10. Along the way, we had JQ, EM, AO, and NF present Exercises 10.9, 10.15, 10.20, and 10.27, respectively.
  • Friday, December 8: Last day! I'll miss this group of students. We continued discussing Chapter 10. DV, EM, JG, and DW presented Problem 10.31, Problem 10.33, Exercise 10.36(a), and Exercise 10.36(b), respectively.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

About This Site

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.