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, January 15: Martin Luther King Jr. Day, no classes.
  • Wednesday, January 17: First day! The first few minutes of class were devoted to me attempting to learn names. Next, I summarized what to expect from the course, toured the course webpage, and summarized a few items on the syllabus. With the time we had left, we briefly tinkered with Spinpossible.
  • Friday, January 19: I think we had a great second day! After fielding a few questions about the syllabus and reminding students about the day-to-day structure, we divided the class up into 10 small groups, each tasked with discussing one of the homework problems. We had RF, RD, SJ, JB, VM, ER, and HH presented their proposed solutions to Problems 2.1, 2.2, 2.3, 2.4, 2.5, 2.7, and 2.8, respectively. Along the way I presented a solution to Problem 2.6. We will wrap up Problems 2.9, 2.10, and 2.11.

Week 2

  • Monday, January 22: Another great day! We had AS, ML, SJ, JW, QC, LD, and MD present Problems 2.10, 2.11, 2.12/2.14, 2.13(a), 2.13(b), 2.14(c), 2.14, and 2.17, respectively. Along the way, I presented Problems 2.9, 2.18, and 2.19. We had 2 minutes at the end to discuss Weekly Homework 1 and do a quick introduction to LaTeX.
  • Wednesday, January 24: After a quick summary of what a group is, we jumped into student presentations. We had TC, ER, JC, AM, JW, FA, and AT present Problems 2.20(a), 2.21(a), 2.22(a), 2.23(a), 2.24, 2.26(a), and 2.26(b), respectively. Along the way, I discussed part (b) for 2.20-2.23. We discussed LaTeX briefly in the last few minutes of class.
  • Friday, January 26: We accomplished a lot today. Most presenters were chosen in advance. We had MM, ML, MS, AS, FZ, BH, and LD present Problem 2.27, Problem 2.28, Theorem 2.29, Problem 2.31, Problem 2.32, Problem 2.34, and Theorem 2.37, respectively. Along the way, I covered Problems 2.35 and 2.36.

Week 3

  • Monday, January 29: After reviewing Problems 2.35, 2.36, and 2.38, we spent 10 minutes discussing LaTeX. Next, I somewhat sloppily divided the class up into 5 small groups, each tasked with discussing one of the homework problems. After a few minutes, we had CW, JB, MS, and RF present Theorem 2.39, Problem 2.40, Corollary 2.41, and Theorem 2.42, respectively. We will discuss Theorem 2.43 on Wednesday.
  • Wednesday, January 31: Even though we didn't get to all the problems today, I feel like we covered a tremendous amount of content. We had JC, DS, RH, GT, DC, and JS present Theorem 2.43 (which as left over from Monday), Theorem 2.44, Theorem 2.45, Theorem 2.47(a), Theorem 2.47(b), and Problem 2.48, respectively. Theorem 2.50 and Problems 2.53-2.55 will likely get discussed on Friday when Dr. Falk covers for me.
  • Friday, February 2: Dr. Falk covered for me while I was out of town. My understanding is that Dr. Falk covered Theorem 2.50, Problem 2.53, and Problem 2.54. Along the way, there was also some discussion of the free group on one letter and the free group on two letters.

Week 4

  • Monday, February 5: We almost got caught up today. After I lead some discussion about Problems 2.55-2.58, we had TC, QC, MM, MD, AT, AM, CW, BH, and RD present Problems 2.59(a), 2.59(b), 2.59(c), 2.61, 2.62, 2.70(a), 2.70(b), and 2.70(c), respectively. We also discussed Problems 2.64-2.66. We will discuss Theorem 2.63, Problem 2.67, and Problem 2.69 next time.
  • Wednesday, February 7: It was a busy day and we managed to get caught up on all the outstanding problems. We had RH, FZ, FA, ML, HH, VM, AS, and SJ present Theorem 2.63, Problem 2.67, Problem 2.69, Problem 2.70(d), Problem 2.70(e), Problem 2.70(f), Problem 2.70(g), and Problem 2.71(ab). In the last couple minutes of class I cranked through Problem 2.71(c).
  • Friday, February 9: The students took the in-class portion of Exam 1.

Week 5

  • Monday, February 12: I lectured over the remaining portion of Chapter 2.
  • Wednesday, February 14: I lectured over the beginning of Chapter 3. We discussed up to Problem 3.12.
  • Friday, February 16: After handing back the in-class portion of Exam 1, we split the class up into several small groups. We had LD, ER, JB, ML, CW, DC, RH, FZ, and RD present Problems 3.13(a), 3.13(b), 3.13(c), 3.13(d), 3.15(a), 3.15(b), 3.16(a), 3.16(b), and 3.16(e), respectively. Along the way, I discussed Problems 3.14 and 3.16(c). We will tackle the leftover problems on Monday.

Week 6

  • Monday, February 19: We spent the first few minutes summarizing the notion of the center of a group. Next, AM, GT, BH, JW, JC, MD, DC, JS, and SJ presented Problem 3.17, Theorem 3.19, Problem 3.20, Theorem 3.21, Problem 3.22(a)(b), Problem 3.22(c)(d), Problem 3.22(e), Problem 3.22(f)(g), and Problem 3.22(j)(k), respectively. Along the way, I presented Problem 3.18. We will address parts (h) and (i) of Problem 3.22 on Wednesday.
  • Wednesday, February 21: After handing back the take-home portion of Exam 1, we jumped right into student presentations. We had QC, RF, VM, RD, AT, JS, MS, and AS present Problem 3.22(h), Problem 3.22(i), Theorem 3.23, Problem 3.25, Problem 3.27, Problem 3.28, Problem 3.29, and Problem 3.30, respectively. Along the way, I also quickly proved Theorem 3.26. We will take care of Problems 3.31 and 3.32 next time.
  • Friday, February 23: We got a lot done today and nearly got through everything. We kicked off with a quick discussion of matching for finite groups involving Cayley diagrams. We had MM, MD, VM, CW, FZ, TC, ER, MS, and SJ present Problem 3.31, Problem 3.32, Problem 3.33, Theorem 3.34, Problem 3.35, Problem 3.36, Problem 3.37, Problem 3.38, and Problem 3.39, respectively. We will take a look at Problem 3.40 on Monday.

Week 7

  • Monday, February 26: We had FA/JC, AT, QC, AM, and SJ present Problems 3.40(b), 3.40(cd), 3.40(e), 3.41, and 3.42. This was followed by a discussion of Problem 3.43, Problem 3.44, Theorem 3.45, and Problem 3.47. There were a few we didn't get to, so we will have to address them next time.
  • Wednesday, February 28: After a quick review of the 3 different ways to view isomorphisms, we jumped into student presentations. We had DC, MS, ML, JB, MD, and RD present Problem 3.48, Problem 3.49(a), Problem 3.49(c), Theorem 3.51, Theorem 3.52, and Theorem 3.53. Along the way, I presented Problem 3.48(b) and Problem 3.50. We didn't get to Theorems 3.54 or 3.55.
  • Friday, March 2: Dr. Falk covered for me while I was out of town. My understanding is that VM presented Theorem 3.54 and then Dr. Falk covered up to Problem 3.59.

Week 8

  • Monday, March 5: We tried to get caught up today, but we are still a bit behind. Before class started, I sketched skeleton proofs for Theorems 3.60-3.62 on the board. After some discussion of a few of the problems from last time, we divided the class up into 6 small groups, each tasked with discussing the proof/solution for a problem or two. We had AM, HH, BH, and JC/VM presented Theorem 3.60, Theorem 3.61, Theorem 3.62, and Problem 4.4, respectively. After Theorem 3.62 was presented, I discussed Problems 3.64 and 3.65.
  • Wednesday, March 7: I think we were super productive today. After discussing the upcoming exam, we jumped right into the current problems. I presented Theorem 4.8, Problem 4.11, Problem 4.12, Problem 4.14, Theorem 4.17, Theorem 4.19, and Corollary 4.20. Along the way, we had LD, JW, and FA present Problem 4.13, Theorem 4.15, and Corollary 4.18, respectively. Theorem 3.63 is still outstanding and we will address it next week.
  • Friday, March 9: The students took the in-class portion of Exam 2.

Week 9

  • Monday, March 12: I spent the entire class period lecturing. We finally wrapped up Theorem 3.63 and then spent a few minutes discussing automorphisms. With the time we had left, we covered Problems 4.21 and 4.22.
  • Wednesday, March 14: I continued lecturing over Chapter 4.
  • Friday, March 16: More lecturing. We more or less covered up to Theorem 4.39.

Week 10

  • Monday, March 26: Final day of lecturing before jumping back into student presentations. We covered Corollary 4.40, Theorem 4.41, Problem 4.42, Corollary 4.43, Problem 4.46, and Corollary 4.47.
  • Wednesday, March 28: After reviewing some of the key ideas from Section 4.1, we had RH, QC, FA, BH, and ER present Theorem 4.44, Theorem 4.45 (forward implication), Theorem 4.45 (punchline for reverse implication), Theorem 4.45 (intermediate set containments), and Problem 4.48, respectively. Part of Theorem 4.44 remains outstanding and I offered up some extra credit for figuring it out.
  • Friday, March 30: Before doing any student presentations, we discussed Corollary 4.50, Problem 4.53, Definition 4.54, Theorems 4.55-4.58, and Problems 4.59 and 4.60. Next, we split the class up into 6 small groups, each tasked with discussing one of the remaining problems. We had SJ, AM, AS, MD, JC, and TC presented Problem 4.65, Problem 4.66, Problem 4.67, Problem 4.68, Problem 4.69, and Theorem 4.70, respectively.

Week 11

  • Monday, April 2: Somehow we managed to get through everything today. After splitting up into a few small groups, we had HH, MM, GT, JB, QC, MD, JC, and RD present Problem 4.73, Problem 4.74, Problem 4.77, Problem 4.80, Problem 4.81, Problem 4.82, Problem 4.83, and Problem 4.84, respectively. This was followed by me presenting the proofs of Theorems 4.78 and 4.79.
  • Wednesday, April 4: Busy day! We had MD, AM, RH, JW, VM, and FA present Problem 4.85(ab), Problem 4.86(cd), Problem 4.87, Problem 4.88, Theorem 4.89, and Corollary 4.90, respectively.
  • Friday, April 6: David Deville covered for me while I was out. RD, QC, FZ, RH, and MD presented Problem 4.94(a), Problem 4.94(b), Problem 4.95(b), Problem 4.96, and Lemma 4.97, respectively.

Week 12

  • Monday, April 9: After discussing my crazy bike race for a bit, we jumped back into mathematics. After I presented the proofs of Theorems 4.97 and 4.106, we had RD, AM, MS, MM, and RH presented Problem 4.101, Problem 4.102, Problem 4.103, Problem 4.107, and Problem 4.110, respectively.
  • Wednesday, April 11: Another busy day. We had ML, AS, ER, RD, GT, MD, JB, and TC presented Problem 4.111, Problem 5.2, Problem 5.3, Problem 5.4, Problem 5.5, Problem 5.6, Problem 5.7, Theorem 5.8, and Problem 5.20, respectively.
  • Friday, April 13: Dr. Falk covered for me while I was at the Southwestern Undergraduate Mathematics Research Conference with several students from class. Dr. Falk lectured over the beginning of Chapter 8.

Week 13

  • Monday, April 16: Exciting and busy day! We more or less wrapped up Chapter 5. We had FA, JW, and JB present Problem 5.24, Problem 5.25, and Theorem 5.30 and I presented Theorem 5.21, Corollary 5.22, Problem 5.28, Theorem 5.29, Problem 5.31, Theorem 5.32, Problem 5.33, and Theorem 5.35 (although, we didn't finish 5.35).
  • Wednesday, April 18: The students took the in-class portion of Exam 3.
  • Friday, April 20: I lectured over the rest of Chapter 5 and half of Chapter 6.

Week 14

  • Monday, April 23: I continued lecturing over Chapter 6.
  • Wednesday, April 25: More lecturing. We wrapped up Chapter 6 and discussed some of Chapter 7.
  • Friday, April 27: We had LD and HH present Theorem 7.12 and Problem 7.18, respectively. Along the way, I presented Problem 7.15, Problem 7.16, Problem 7.17, and most of the proof of the Theorem 7.20 (First Isomorphism Theorem).

Week 15

  • Monday, April 30: After wrapping up the proof of the First Isomorphism Theorem, we had MS, LD, RF, and AS present Problem 7.21, Problem 7.23, Problem 8.9, and Problem 8.15, respectively. We'll tackle the ones we didn't get to next time.
  • Wednesday, May 2: We didn't get through everything today, but we accomplished a lot. We had JS, ML, MM, ER, TC, BH, JB and LD present Problem 8.16, Problem 8.17, Theorem 8.18, Problem 8.31(a), Problem 8.31(b), Problem 8.31(c), Problem 8.32, and Theorem 8.38(b), respectively. Along the way, we also addressed Problem 8.20.
  • Friday, May 4: Last day of classes! I had a fun semester with this group of students. We had AM, GT, FZ, LD, JW, and RD present Problem 8.33, Problem 8.36(a), Problem 8.36(b), Theorem 8.39, Theorem 8.42, and Theorem 8.43, respectively. Along the way, I presented Problem 8.48.

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.