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 10: First day! After attempting to learn names, I gave a very quick tour of the course webpage and discussed in general terms what a typical class meeting would look like. Next, I gave a brief summary of what "abstract algebra" is and then jumped into Chapter 2 and started discussing Spinpossible.
  • Wednesday, January 12: Great first day of presentations! After addressing a few questions about the syllabus, we divided the class up into several small groups and then had SR/KB/KR, NH, AC, JO, and JI present Problems 2.2-2.6, respectively. We will catch up on Problem 2.7 next time.
  • Friday, January 14: We had folks volunteer for presentations today. We had HP, RW, and AH present Problems 2.7, 2.8, and 2.11, respectively. Along the way, I presented Problems 2.9 and 2.10.

Week 2

  • Monday, January 17: MLK Day! No classes.
  • Wednesday, January 19: Lots of great discussion today! We had SR, KB, DG, HJ, and PV present Problems 2.12, 2.13(a), 2.13(b), 2.14, and 2.17, respectively. I also squeezed in some discussion of Problems 2.18 and 2.19. We'll try to get caught up next time.
  • Friday, January 21: Another great day! We had KF, TR, NH, MW, and MT present Problems 2.20, 2.21, 2.22, 2.23, and 2.24, respectively.

Week 3

  • Monday, January 24: We got caught up! We had ND, MH, PB, and BH present Problem 2.26(b), Problem 2.27, Theorem 2.29, and Problem 2.32, respectively. Along the way, we also discussed Problems 2.26(a), 2.28, 2.31(a), and 2.31(b).
  • Wednesday, January 26: After reviewing the defintion of a group, we covered Problem 2.34, Problem 2.35, Problem 2.36, Theorem 2.37, and Problem 2.38. KC and JO presented Problem 2.34 and Theorem 2.37, respectively, while I led discussion on the rest.
  • Friday, January 28: We had SR, MB, KR/TR, and TB presented Theorem 2.39, Theorem 2.41, Theorem 2.42, and Theorem 2.43, respectively.

Week 4

  • Monday, January 31: Well, we got behind again. But I think it is worth taking our time on some of these problems. We had SC, LS, and HJ present Theorems 2.44, 2.47(a), and 2.47(b), respectively. I also discussed Theorem 2.47(c) and Problem 2.48.
  • Wednesday, February 2: We had JB, JI, and KB present Theorem 2.52, Problem 2.55, and Problem 2.56, respectively. We also squeezed in a discussion of Problem 2.60.
  • Friday, February 4: We had MT, JC, JO, and TH present Problems 2.57, 2.59, 2.62, and 2.64, respectively. We also discussed Problems 2.58, 2.61, and 2.63.

Week 5

  • Monday, Febuary 7: Productive day. We had ND, KF, LB, LS, and SR present Theorem 2.65, Problem 2.68, Problem 2.69, Problem 2.71(abc), and Problem 2.71(def), respectively.
  • Wednesday, February 9: We kicked off with reviewing on the Cayley diagrams in Problem 2.72 and then AH presented Problem 2.73. With the time we had left, we discussed Problem 2.75, Problem 2.76, and Theorem 2.77.
  • Friday, February 11: Wildly productive day! We had JB/SR, BH, PB, KF, KC, and MW present Theorem 2.78, Problem 2.79, Theorem 2.80, Problem 2.82(a), Problem 2.82(b), and Problem 2.83(a), respectively.

Week 6

  • Monday, Febuary 14: After MT revisited Problem 2.83(b), we cranked through Problems 3.1-3.3, and then MH presented Problem 3.5. With the time left, we discussed the forward implication of Theorem 3.6.
  • Wednesday, February 16: We had SR, RW, KB, and TR present Theorem 3.6, Theorem 3.10, Problem 3.12, and Problem 3.13, respectively. Along the way, I discussed 3.7-3.9.
  • Friday, February 18: We covered a lot of ground today. After TR wrapped up Problem 3.13, I quickly marched us through discussions of Problems 3.16, 3.18, 3.19, and 3.20. Next we had KD, TB, JC, SC, and PV present Problems 3.14(ab), 3.14(cd), 3.15(abcde), 3.15(fghi), and 3.17(b), respectively.

Week 7

  • Monday, Febuary 21: After discussing the upcoming exam, we jumped into discussing problems. I presented Theorem 3.21, Theorem 3.23, and part of Problem 3.24. Along the way, KD presented Problem 3.22.
  • Wednesday, February 23: Roughly half the students took the in-class portion of Exam 1.
  • Friday, February 25: Due to the weather on Wednesday, the rest of the class took Exam 1.

Week 8

  • Monday, Febuary 28: After handing back the in-class exams, I briefly discussed a few of the problems from the exam. With the time we had left, we worked through computing the centers for $D_4$ and $Q_8$ and then started discussing subgroup lattices.
  • Wednesday, March 2: We kicked off by discussing the solution to Problem 3 on the in-class portion of Exam 1. Next, we cranked through the construction of several subgroup lattices and then discussed Theorem 3.26, Problem 3.27, and Theorem 3.28.
  • Friday, March 4: I continued lecturing. We tinkered with a couple additional subgroup lattices and then started discussing isomorophisms and matchings of Cayley diagrams.

Week 9

  • Monday, March 7: After a discussion of matching Cayley diagrams versus matching subgroup lattices, we had ND, KC, SC, and MB present Problems 3.32, 3.39, 3.40, and 3.41, respectively.
  • Wednesday, March 9: We had MW, SR, and KR present Problems 3.43, 3.50, and 3.51(a), respectively. With the time we had left, we discussed the three ways to verify whether two groups are isomorphic and cranked through Problem 3.49.
  • Friday, March 11: After revisiting Problem 3.51(a), we had TH, JI, and SR present Problem 3.51(b), Problem 3.51(c), and Problem 3.52, respectively. Next, we discussed Theorems 3.53 and 3.54. We will get caught up on the stuff we didn't get to after spring break.

Week 10

  • Monday, March 21: We discussed 3.54-3.60. Along the way, MT presented a portion of Theorem 3.57.
  • Wednesday, March 23: Exciting day! At least I was excited;) Together we discussed 3.61-3.65, 3.66(b). Along the way, I made regular references to analogous situtations in linear algebra. With the few mintues we had left, KC presented Problem 3.66(a).
  • Friday, March 25: Ian Williams covered for me while I was out of town. NH and MH presented Theorems 3.65 and 3.67, respectively, and then Ian led discussions on 4.1-4.4.

Week 11

  • Monday, March 28: I lectured my ass off today! Roughly, we covered 4.5-4.21.
  • Wednesday, March 30: More lecturing, even though that wasn't the plan. We discussed 4.24-4.35.
  • Friday, April 1: We had KR, JO, AH, and MT present Problem 4.36, Problem 4.37, Problem 4.38, and Theorem 4.41, respectively.

Week 12

  • Monday, April 4: We had HP, LS, SR, and DG present Problems 4.42, 4.46, 4.47, and 4.49, respectively. Along the way, we briefly discussed Corollary 4.43 and Theorem 4.44 and got started on a proof for Theorem 4.45.
  • Wednesday, April 6: The students took the in-class portion of Exam 2.
  • Friday, April 8: I wrapped up the proof of Theorem 4.44 and then discussed cosets and introduced the quotient process. We tinkered with several examples.

Week 13

  • Monday, April 11: We started discussing the symmetric group. We more or less covered 4.67-4.87.
  • Wednesday, April 13: More lecturing. We covered the rest of Section 4.3.
  • Friday, April 15: We reviewed a few key ideas from Section 4.3 and then did a quick summary of Cayley's Theorem. With the few minutes we had left at the end of class, we introduced the parity of a permutation and started discussing the alternating group.

Week 14

  • Monday, April 18: After splitting the class up into small groups, we had DG/MH, HJ/ND, JC, and SR/RW present Problem 4.115, Problem 4.116, Problem 4.117, and Theorem 4.119, respectively.
  • Wednesday, April 20: Very productive day! We had MB, JB, LS, AH, KF,, TH, and ND presented Problems 4.120, 4.125, 6.26, 6.27, 6.28(a), 6.28(b), 6.29(a), and 6.29(b), respectively.
  • Friday, April 22: It was mostly me lecturing today. I reviewed the big picture of normal subgroups and quotient groups. Along the way, we discussed 6.31, 6.32, 6.33, 6.34, and 6.36.

Week 15

  • Monday, April 25: We jumped into Chapter 7 and revisited the basics of homomorphisms. The bulk of class time was spent on discussing Figure 7.2 and proving Theorem 7.13.
  • Wednesday, April 27: We reviewed Figure 7.2 and Theorem 7.13 and then KD and KB presented Problem 7.14 and Theorem 7.15. After that I proved the First Isomorphism Theorem.
  • Friday, April 29: Our last day of class! And it was another productive day. After reviewing the First Isomorphism Theorem, I presented Problem 7.23 and then LS, HP, and MB presented Problems 7.24, 7.25, and 7.26, respectively. With the time we had left, I reviewed some fundamental defintions related to rings. Thanks for a great semester!

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.