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

• Wednesday, January 20: First day! After attempting to learn names, we launched into 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.
• Friday, January 22: Great first day of student presentations! I was extremely impressed that so many people readily volunteered. After answering a few quick questions about the syllabus, I spent a few minutes highlighting a few key features of Spinpossible. Next, we had JK, GG, LF, IA, EO, KS, and ES present 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, and 2.9, respectively. I was really pleased with how the presentations went, both for the presenters and for the audience. We didn't get to 2.10-2.12. We might come back to these on Monday.

## Week 2

• Monday, January 25: Today was a busy day. As is typical at this point in the course notes, confusion and understanding seem to wax and wane, but I'm not at all worried about it. After a quick discussion about possible ambiguities in our intuitive rules for a group, we had AM2, HG, CS, MT, SW, SC, RW, and AM1 present 2.10 (outstanding from last time), 2.13, 2.15, 2.16/2.17, 2.18, 2.19, 2.21, and 2.22, respectively. With the few minutes of class we had remaining, I discussed 2.20 and 2.23.
• Wednesday, January 27: Once again we didn't get through all the problems, but I felt like we covered quite a bit of material. The first 15 minutes or so was devoted to me summarizing a few key ideas about Cayley diagrams. I also discussed problems 2.24, 2.25, 3.4, and 3.7. Next, we had AW, CL, NW, and EO presented 3.1, 3.2/3.3, 3.5/3.6, and 3.8, respectively. We'll have AH and MT present 3.9 and 3.10, respectively.
• Friday, January 29: The first half of class was devoted to introducing the use of LaTeX. The rest of our class session was devoted to AH, MT, and AT presenting 3.9, 3.10, and 3.11, respectively. Along the way, I provided alternate drawing of the Cayley diagram for Exercise 3.11, so that we could compare to AT's version.

## Week 3

• Monday, February 1: Today was a good day. We had KF, AW, BS, JK, CM, and RV present 3.12, 3.13, 3.14, 3.15, 3.16, and 3.17, respectively. Exercise 3.14 is an important one to keep in mind.
• Wednesday, February 3: I had my doubts about getting through everything today. We came really close. All of today's problems were related to the concept of subgroup. We had KG, CS, ER, BA, CL, IA, and AM1 present 4.3, 4.5, 4.6/4.7/4.8, 4.10, 4.11, 4.12, and 4.13/4.14, respectively. We didn't get to 4.15, which KS will kick off with next time.
• Friday, February 5: Today's problems focused on understanding when two groups are isomorphic. The problems presented today are among my favorite in the course. It was an action-packed day with KS, MT, AH, ES, SC, MN, AM2, and EO presenting 4.15, 4.16, 4.19, 4.20, 4.23/4.24, 4.25, and 4.26, respectively. We didn't get to 4.27/4.28. MT will kick off with those on Monday.

## Week 4

• Monday, February 8: The major theme of the day was binary operations. After a short discussion of where we are headed, MT presented 4.27/4.28. Next, we had KG, KW, RV, RW, AM1, and EO presented 5.8, 5.9, 5.10, 5.11, 5.13, and 5.14, respectively. EO's attempt at Theorem 5.14 was pretty close, but had some minor notational issues. Nonetheless, I was impressed.
• Wednesday, February 10: The first few minutes of class were devoted to discussing the upcoming exam. Next, we looked at some interesting errors on student work from Weekly Homework 1. This took quite a bit of time, but I think it was worthwhile. With the time we had left, BA, HG, and FT presented 5.16, 5.17, and 5.20, respectively. There were a bunch of problems that we didn't get to. Andrew will be covering for me on Friday and he will pick up where we left off.
• Friday, February 12: Andrew covered for me while I was out of town. According to his notes, MN, SW, LF, NW, ER, BS, EO, and GG presented 5.21, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28, and 5.29, respectively.

## Week 5

• Monday, February 15: We chatted very briefly about the exam and then I spent a few minutes comparing solving equations in linear algebra to solving equations in the context of groups. Next, we had CM, FT, AT, KF, RV, and KS present 5.30, 5.32, 5.33, 5.34, 5.35, and 5.37, respectively. There were a few problems we didn't get to and students do not need to worry about these for the exams.
• Wednesday, February 17: The students took the in-class portion of Exam 1.
• Friday, February 12: After a quick discussion of the take-home portion of Exam 1, we took care of a few outstanding problems. We had ES, SC, and IA present 5.38, 5.40, and 5.41, respectively. Next, I spent a few minutes summarizing "the taxonomy of groups" and the phrase "up to isomorphism." I then lectured over 5.39, 5.42, 5.43, and 5.45. I'll continue lecturing until the take-home exam is due next week.

## Week 6

• Monday, February 22: After handing back the in-class portion of Exam 1, I launched into lecturing. We picked up where we left off on Friday with a discussion of the taxonomy of groups and then continued on with the course notes. I covered 5.46-5.51.
• Wednesday, February 24: We picked up where we left off. Theorem 5.53 was on Exam 1, so I talked about it very briefly. Next, we carefully worked through 5.55-5.58. We left 5.55(d) open with the intention of discussing it later.
• Friday, February 26: New room! We moved into a room with more white board space, which makes me very happy. I divided the class up into 6 small groups and each group was tasked with discussing one of 5.61, 5.62/5.67, 5.63, 5.64, 5.66, and 5.68. For 10-15 minutes, I let each group discuss their proposed solution to the problem they were assigned. Next, each group's spokesperson discussed their group's solution. We had AM2, BS, CS, and HG present 5.61, 5.62/5.67, 5.63, and 5.64, respectively. CS's presentation of Theorem 5.63 was excellent. We ran out of time and didn't get to discuss 5.66 and 5.68. We'll likely skip discussing 5.68, but we will squeeze in 5.66 next time.

## Week 7

• Monday, February 29: I was a little out of it today, but had a lot of fun nonetheless. Just like last time, we divided the class up into several small groups. After the groups had a chance to work on their respective problems, I finally got around to discussing 5.56(d). Next, KS walked us through 5.66, which was left over from last time. The rest of the problems focused on subgroup lattices. Problems 5.69, 5.70, 5.71, 5.72, 5.73, 5.74, and 5.76/5.77 were presented by GG, MT, SC, EO, RV, MT, and IA, respectively. Along the way, I discussed 5.75. Good day!
• Wednesday, March 2: Class began with me summarizing the homomorphic property and what it means. Next, we divvied up the class into several small groups to discuss problems from the homework. After a few minutes we started discussing each group's proposed solution. We had AW, AH, CL, ES, and CM present 5.80, 5.81(a), 5.81(bc), 5.82, and 5.83, respectively. We didn't get through all of the problems, so we will start with those next time.
• Friday, March 4: We picked up where we left off. LF, MN, RW, BA, and IA presented 5.84, 8.85, 5.87, and 5.89, respectively. Along the way, I discussed 5.88. This ticked off all the leftovers from last time and got us most of the way through the new problems. I will discuss Problem 5.91 at the beginning of class next time. We may end up skipping a discussion on Theorem 5.90.

## Week 8

• Monday, March 7: Today began with me discussing which theorems at the end of Chapter 5 required the function to be an isomorphism. We will return to this issue in a couple chapters. Next, we split the class up into seven small groups, each of which was tasked with discussing a particular homework problem. ER, GG and KF, AT, and AM1 presented 6.3/6.4(abcdef), 6.4(ghijk)/6.5, 6.6, and 6.7, respectively. Unfortunately, we didn't get to 6.8, 6.9, 6.10, and 6.11, which we will try to squeeze in next time.
• Wednesday, March 9: JK, AW, SC, ES, IA, and LF presented 6.8(b), 6.10, 6.11, 6.14, 6.16/6.17, and 6.18, respectively. We'll do with 6.8(a) and 6.20 at the beginning of class next time.
• Friday, March 11: We finally got to Theorem 6.8(a), which SW presented. EO also presented 6.20, which was another left over. After doing these problems, we split up into small groups to tackle most of the problems from homework. We had MT, RV, RW, AM2, MT, and BS present 6.21, 6.23, 6.28, 6.29, 6.30, and 6.31, respectively. We'd handle 6.24-6.27 after spring break.

## Week 9

• Monday, March 21: I spent the whole class lecturing. I picked up where we left off and covered material in the notes up to and including Corollary 6.36. We started discussing Theorem 6.37 but ran out of time.
• Wednesday, March 23: After a quick chat about the upcoming exam, we divvied the class up into small groups to work on problems from the homework. When we brought everyone back together, we had AM2, MN, KF, AT, NW, KG present 6.37, 6.38, 6.41, 6.40(a), 6.40(b), and 6.43, respectively. Along the way, I discussed 6.39 and 6.42. We'll wrap up 6.45, 6.46, 6.47, and 6.48 next time.
• Friday, March 25: It was a fairly busy and action-packed class. After some discussion about moving the exam to Wednesday, we jumped right into tackling the problems that were left over from last time. After an involved discussion about 6.45, CS and AH presented 6.46 and 6.47, respectively. Then we quickly dispensed with 6.48. Next, we cranked through all of Section 6.2. We had MT, CM, IA, JK, AM, RV, and SC present 6.50, 6.51, 6.52, 6.53(a), 6.53(b), 6.53(c), and 6.54, respectively.

## Week 10

• Monday, March 28: As usual, we split up into several small groups and after a short period of time, we regrouped to discuss what each group came up with. We had NW, CL, KG, KS, CM, SC, MT, ER, AT, and BA present 6.56, 6.57, 6.59, 6.60, 6.61, 6.62, 6.63, 6.64, 6.65, and 6.66, respectively. This was a nice set of problems to do just before the exam. It seemed like no one had any difficulty.
• Wednesday, March 30: The students took Exam 2.
• Friday, April 1: I lectured over Section 6.3. In particular, I discussed 6.67, 6.68, 6.69, 6.70, 6.76, and 6.77.

## Week 11

• Monday, April 4: I continued lecturing over Section 6.3 of the course notes. We covered 6.71-6.75, 6.78, 6.79.
• Wednesday, April 6: We accomplished a lot today! We had EO, CS, SC, AM1, ER, ES, and IA present 6.80, 6.81, 6.82, 6.83, 6.84, 6.85, and 6.86, respectively.
• Friday, April 8: Another busy day. We had AW, KG, MT, MN, CL, RW, AH, and BA present 6.92, 6.95, 6.96/6.97, 6.100, 6.101, 6.102, 6.103, 6.104, respectively. As a class, we also discussed 6.98 and 6.105. As promised, I will send you a write up of the proof of Lemma 6.91.

## Week 12

• Monday, April 11: After some discussion of Exam 2 and grades, I talked for a bit about cosets. Along the way, I completed 7.2, 7.3, 7.5, 7.6, and 7.7. We also had SC and ES show us 7.4(ab) and 7.4(cd), respectively.
• Wednesday, April 13: After some discussion of cosets and Lagrange's Theorem, we launched into small groups discussing problems. We had KS, CL, IA, BS, MT, GG, RV, and AM2 present 7.8(1), 7.8(2), 7.8(3), 7.8(4), 7.18, 7.19, 7.21, and 7.22, respectively. This is hard stuff and I'm really pleased with how things are going.
• Friday, April 15: We had SW, NW, LF, BA, AW, CM, and AM1 present 7.25, 7.26(a), 7.26(b), 7.27, 7.29, 7.30, and 7.31, respectively. We didn't quite finish 7.31, so we'll start with that next time.

## Week 13

• Monday, April 18: We kicked off with me walking through the proof of Theorem 7.31, which provides an alternative for checking whether a subgroup is normal. Next, we split into small groups to discuss most of the problems due today. We had MN, KF, AT, AH, and MT share their group's proposed solutions to 8.6, 8.7, 8.8, 8.13, and 8.15, respectively. We still have 8.10, 8.11, 8.16, 8.17, and 8.18 as outstanding.
• Wednesday, April 20: I was really impressed with how things went today. After some discussion of 8.11 and 8.16, we split up into small groups as usual. After a few minutes, we had ER, KF, HG, ES, EO, LF, and GG share solutions to 8.17, 8.18, 8.19, 8.20, 8.21, 8.22, and 8.25, respectively. Theorem 8.10 is still outstanding, but we will tackle it on Friday.
• Friday, April 22: We finally wrote down a proof of Theorem 8.10 and then I lectured over quotient groups.

## Week 14

• Monday, April 25: The students took the in-class portion of Exam 3.
• Wednesday, April 27: At lightning speed, I lectured over Chapter 9: Homomorphisms and the First Isomorphism Theorems.
• Friday, April 29: One of our best days! AM2, GG, EO, AM1, and CS presented 9.14/9.21, 9.15, 9.16, 9.22, and 9.23, respectively. There are a few problems we didn't get to, but the conversations we had on the ones we did definitely made up for it.

## Week 15

• Monday, May 2: Our first day of discussing rings. We were a bit all over the place, but I think folks learned a few things nonetheless. We had CS, AH, SW, and JK discuss 10.8/10.15/10.16, 10.10, 10.17, and 10.18 respectively.
• Wednesday, May 4: After handing back the take-home portion of Exam 3, I lectured over 10.10, 10.25, 10.31, and 10.37. We will do more of the same next time.
• Friday, May 6: Last day of class! I'm going to miss this group of students. After spending a little bit of time reflecting on the semester, I lectured (rather quickly) over the remaining content in Chapter 10.

# Dana C. Ernst

Mathematics & Teaching

Northern Arizona University
Flagstaff, AZ
Website
928.523.6852
Instagram
Strava
GitHub
arXiv
ResearchGate
Mendeley
Impact Story
ORCID