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 29: First day! After attempting to learn names, we launched in the Setting the Stage activity. I was impressed by the level of engagement and everyone's willingness to share. Some wonderful ideas were discussed! The last bit of class was devoted to touring the course website and discussing a few important items from the syllabus.
  • Wednesday, August 31: Absolutely spectacular first day of presentations. I was extremely impressed that so many people readily volunteered. After dividing the class up into several small groups to discuss the homework problems, we had LL, AT, JM, JS, AO, EB, SS1, and MR present 2.2, 2.4, 2.5, 2.6, 2.8, 2.9, 2.10, and 2.13, respectively. Along the way, we also discussed 2.3 and 2.12 as a class.
  • Friday, September 2: This might be the best start to the semester that I've ever experienced. Awesome discussions so far. After answering a few quick questions, we split up into small groups just like last time. We had DJ, SC, KS, SS1, and KE present 2.16/2.17, 2.18, 2.20, 2.21, and 2.22, respectively. Along the way, I quickly discussed 2.19. We were short on time at the end, so I very quickly summarized 2.24 and 2.25. We never go to 3.1-3.3, so we will start with those next time.

Week 2

  • Monday, September 5: Labor Day. No class.
  • Wednesday, September 7: We got a lot done today and all the discussion was productive. We kicked off with me explaining 3.1 and 3.2, which were left over from last time, and I also quickly explained 3.4. After that we split up into small groups to discuss the remaining problems. LG, SS1, JC, BG, JH, and PM present 3.5/3.6, 3.7, 3.8, 3.9(c)/3.10(cd), 3.11(g), and 3.12(g), respectively.
  • Friday, September 9: I spent the first 25 minutes or so discussing the use of LaTeX. Remember to holler if you have questions. The rest of class was spent discussing 3.13 and 3.14, which SM and DJ presented, respectively. If needed, we will revisit 3.15, 3.16, and 3.17.

Week 3

  • Monday, September 12: I was out of town and David Deville covered for me. According to David, here is what transpired. JS, LG, JC, MR, EB, LL, and BG presented 4.2/4.3, 4.5, 4.6, 4.7, 4.8, 4.10, and 4.11, respectively. We should have time to hit up 4.12-4.14 next time.
  • Wednesday, September 14: Today was a recalibration day. I spent the entire time lecturing over 4.12-4.15. We kick off Friday by finishing up discussion of 4.19-4.22.
  • Friday, September 16: Great day and one of my favorite topics! After a quick summary of how to verify that two groups are or are not isomorphic using Cayley diagrams, we split up into 6 small groups. I presented 4.19 and 4.20 and then we had DZ, HR, KE, SM, AO, and ZP discuss 4.21, 4.22, 4.23, 4.24, 4.25, and 4.26, respectively.

Week 4

  • Monday, September 19: We got a lot accomplished today. I spent the first few minutes discussing some thoughts about the first Weekly Homework assignment and then we got down to business. We had JM, AO, AT, AN, SC, and HR present 4.27/4.28, 5.9, 5.10, 5.11, 5.13, 5.14, and 5.16, respectively. Along the way I presented 5.8 and then we wrapped up with the proof of Theorem 5.14 (function composition is associative).
  • Wednesday, September 21: We kicked off with a whole class discussion of 5.20, 5.21, 5.22, and 5.23. These problems were aimed at developing intuition about the formal definition of a group. Next, we had people work in groups of 2-3 on coming up with a proof for Theorem 5.24, which says that the identity in a group is unique. After about 5 minutes, we had JM, SS2, and PM share 3 different proposed proofs.
  • Friday, September 23: Today is a perfect example of why I use to implement an IBL approach in this course. Awesome discussion and many light bulbs getting turned on! We had JM, KE, KS, JK, AN, JH, and DZ present 5.25, 5.26, 5.27, 5.28, 5.30, 5.32(a), and 5.32(b), respectively. With the few minutes we had left at the end of class, we had quick group discussion of 5.33 and 5.34.

Week 5

  • Monday, September 26: Another highly productive day. After discussing the upcoming exam, we got down to business. We started by me discussing 5.36, 5.39, and 5.40. Then we broke up into 6 small groups to discuss most of the remaining problems. We had JK, AN, MR, LL, PM, and SC present 5.35, 5.37, 5.38, 5.41, 5.43, and 5.44, respectively. Along the way, I also addressed 5.42 and discussed the phrase "up to isomorphism."
  • Wednesday, September 28: The students took the in-class portion of Exam 1.
  • Friday, September 30: Since the students are working on the take-home portion of Exam 1, I spent the day lecturing. We began by reviewing the groups we are familiar with and organized them by order and identified which ones were isomorphic to others on our list. Next, we had lengthy discussion of Problem 5.45, which classifies groups of order 4. With the time we had left, we briefly discussed 5.46 and 5.47, which addresses one of the flaws with our original set of 4 rules that defined a group.

Week 6

  • Monday, October 3: After handing back the in-class portion of Exam 1 (I'm happy with how everyone did, by the way!), we picked up where we left off last week. When all was said and done, we discussed 5.46-5.55.
  • Wednesday, October 5: The energy level was a bit low today, but I'm really happy with how things went. SS1, LL, DJ, JM, HR, EB, and SC present 5.56(b), 5.56(c), 5.57, 5.61, 5.62, 5.63, and 5.64, respectively. Along the way, I quickly addressed 5.58.
  • Friday, October 7: Today was a good example of where we tried to do too much. As a result, I think we missed out on an opportunity to resolve some confusion about subgroup lattices. We had AO, JH, KS, DZ, JC, MR, and SS2 present 5.67, 5.68, 5.69, 5.70, 5.71, 5.72, and 5.73, respectively. We will review a couple of these next time and hit up Theorem 5.66 and tackle Exercise 5.74 and Problem 5.75.

Week 7

  • Monday, October 10: After proving Theorem 5.66 as a class, AT did a quick presentation of 5.74, which was left over from last time. Then we quickly discussed 5.75 and 5.78. The remainder of class was spent discussing the formal definition of isomorphism and the homomorphic property. We wrapped up class by doing 5.81 as a class.
  • Wednesday, October 12: We cranked out a bunch of proofs today. I was really happy with the proofs that were presented, as well as the discussion around each problem. We had LL, PM, AT, JS, BG, and LG present 5.80, 5.82, 5.83, 5.84, 5.86, and 5.87, respectively.
  • Friday, October 14: Another action packed day, but I think things went well. We had SM, JH, KS, DZ, JK, and KE presented 5.88, 5.89, 5.90, 5.91, 6.1, and 6.4, respectively.

Week 8

  • Monday, October 17: After groups had an opportunity to discuss problems, we had MR, JK, EB, and JC present 6.5/6.6, 6.11, 6.12, and 6.14, respectively. With the time we had left, we had JM and KS/JS discuss 6.8 and 6.9(a), respectively. Theorem 6.9(b) and Corollary 6.10 are still open.
  • Wednesday, October 19: Today went way better than expected. We had SC/KE, LG, BG, DJ, SM, AN, and SS1 present 6.15, 6.16, 6.17/6.19, 6.20, 6.21, and 6.23, respectively.
  • Friday, October 21: After discussing 6.26 and 6.27 together, we had HR, AN, PM, JH, SS2, BG, and JK present 6.28, 6.29, 6.30, 6.31, 6.32, 6.33, and 6.36, respectively. We will kick off with 6.35 next time.

Week 9

  • Monday, October 24: We kicked off with me proving Theorem 6.35. Next, AN , JS, SS1, and HR tackled 6.38, 6.39, 6.41, and 6.42, respectively. Along the way, KS attempted to address a question in the notes prior to Theorem 6.40. We did not get to 6.37, 6.40, 6.43, and 6.44. We'll address these on Friday.
  • Wednesday, October 26: The students took the in-class portion of Exam 2.
  • Friday, October 28: After handing back the in-class portion of Exam 2, I picked up where we left off in the notes and lectured over some new material. We covered 6.37, 6.40, and 6.44, which were left over from Monday.

Week 10

  • Monday, October 31: I continued lecturing over Chapter 6. We covered 6.45, 6.48, 6.50, 6.51, and most of 6.52. I'll wrap up the last bit of 6.52 next time.
  • Wednesday, November 2: We kicked off by finishing up Theorem 6.52 and then sketched proofs for all four parts of Theorem 6.53. From there we moved into Section 6.3. We had AT, AN, BG, JC, and SC present 6.56/6.57, 6.59, 6.60, 6.61, and 6.63, respectively. With the time we had left, we discussed 6.54 and 6.64.
  • Friday, November 4: After a quick review of 6.64, we split up into small groups as usual. We had EB, LL, LG, MR, SM, PM, and KS present 6.66, 6.67, 6.68, 6.69, 6.70, 6.71, and 6.72, respectively. Along the way, we addressed 6.55.

Week 11

  • Monday, November 7: After handing back the take-home portion of Exam 2, we divided up into small groups to tackle the day's problems. SS2, AT, HR, DZ, SM, and AN presented 6.74, 7.73/6.75, 6.77, 6.80, and 6.81, respectively. Officially, 6.81 didn't get presented by AN, but the answer was correct and I gave her credit for it.
  • Wednesday, November 9: Despite the somber emotional energy in the room post election night, today was a good day. We had PM, LL, SS1, MR, AN, KE, and JS present 6.82, 6.83(a), 6.83(b), 6.84, 6.85, 6.86, and 6.90, respectively. Along the way, we discussed how 6.87 follows from 6.85 and 6.86 and how 6.88 follows from 6.87.
  • Friday, November 11: Veteran's Day! No class.

Week 12

  • Monday, November 14: We spent a good chunk of time proving Lemma 6.91. After that, BG volunteered to prove Theorem 6.92. With the time we had left, we discussed 6.96, 6.97, and 6.98 as a class. We also briefly sketched a proof of Theorem 6.100.
  • Wednesday, November 16: The first few minutes was devoted to discussing cosets. In particular, we discussed 7.5. After that we had LG, DZ, MR, JH, EB, and JK present 6.100, 6.101, 6.104, 6.105, 7.2, and 7.3, respectively. I gave AT credit for 7.4 even though we ran out of time.
  • Friday, November 18: Despite several absences, today was a great day and is another illustration of why I truly believe in the IBL framework. Awesome stuff! We had BG, JH, JS, JC, JK, LG, and PM present 7.6, 7.7, 7.8(1), 7.8(2), 7.8(3), 7.8(4), 7.18, and 7.21/7.22, respectively. Along the way, we discussed 7.15 and 7.19.

Week 13

  • Monday, November 21: After discussing 7.24 and 7.25, we had LL, JC, DZ, EB, SS2, and KE present 7.26, 7.27, 7.28, 7.29, 7.30, and 7.31, respectively. We still need to wrap up the second direction of 7.31 and we skipped discussing 7.33-7.36.
  • Wednesday, November 23: I was impressed with the attendance today. Thanks for showing up the day before Thanksgiving! We had HR, KE, JK, SM, AT, and PM present 8.6, 8.7, 8.13, 8.16, 8.17, and 8.18, respectively. Along the way, we discussed 8.8, 8.10 (although, we didn't have time to write down its proof), and 8.11. Next time, we will prove 8.10 and to discuss 8.19 and 8.20.
  • Friday, November 25: No classes, Thanksgiving Holiday!

Week 14

  • Monday, November 28: The first third of class was devoted to me writing down the proof of Theorem 8.10. After that we had small groups work on 8.19, 8.20, 8.25, 8.26, 8.27, and 8.28. However, we only had time for LL and KE to present 8.19 and 8.20, respectively. We'll tackle the remaining ones after the exam.
  • Wednesday, November 30: Students took the in-class portion of Exam 3.
  • Friday, December 2: I lectured my butt off today. We more or less covered 8.24-8.35.

Week 15

  • Monday, December 5: After returning the in-class portion of Exam 3, I summarized the solution to a couple of the problems. Next, I provided a few hints on problems for the take-home portion of the exam. With the time we had left, I lectured over material from Chapter 9.
  • Wednesday, December 7: More action-packed lecturing. Other than the proof of the First Isomorphism Theorem, we wrapped up Chapter 9. In addition, we covered a substantial chunk of Chapter 10. We will prove the First Isomorphism Theorem on Friday and do some more of the content in Chapter 10.
  • Friday, December 9: Last day of class! I'm going to miss this group of students. Everyone has consistently had a positive attitude. Today, I reviewed a couple homework problems that utilized the First Isomorphism Theorem for groups and then continued lecturing over Chapter 10.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

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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.