#### Abstract Algebra

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, August 28:**First day! I spent most of the time discussing the nuts and bolts of the course. I finished by discussing the Rubik’s Cube.**Friday, August 30:**We started with a few announcements and reminders. Then we jumped into the first Daily Query. SF was the lucky first chosen, but her question wasn’t to bad. I asked her what she did this summer. Next up was KS, DJ, and RZ. Their questions were a little more difficult, but not too bad. I asked KS to name something important/interesting from the syllabus. DJ was asked to mention something from Chapter 1: Introduction. Lastly, RZ was asked to state any theorem from Chapter 2: Preliminaries. I’m hoping that people’s nerves are more at ease. After the Query, we jumped into student presentations. DT was extremely brave and stepped up to present 2.12. She did a great job, especially for a first go. We spent a good chunk of time discussing ways to improve DT’s proof. Next, MH and TL each presented half of part 1 of 2.16. Despite the fact that we didn’t cover as much as I’d hoped, I’m extremely pleased with how the first day of presentations went.

## Week 2

**Monday, September 2:**Labor Day. No class!**Wednesday, September 4:**Very productive day. In the interest of time, I was going to skip the Daily Query, but PT “insisted” that we do it, so I asked him a general question about equivalence relations, which he answer satisfactorily. Next, we had a bunch of people simultaneously put up proposed solutions to several exercises. After everyone was done writing, we discussed each of the students as a class. Here is who did what: WW (2.22), RG (2.23), TF (a third of 2.39), SA (a third of 2.39), JR (a third of 2.39, a fourth of 2.68), DJ (2.40, a fourth of 2.68), SM (2.43), SB (2.50), PT (half of 2.61), TJ (half of 2.61), SF (a fourth of 2.68), and MH (a fourth of 2.68).**Friday, September 6:**I thought today was awesome. We started with the Daily Query. First, we had JR2 explain to use what a corollary was. Next, RG explained what definitions are and why they are important. After that MH discussed what lemmas are. For the last query, DT had to address the question, “Is a rectangle a trapezoid?” This spawned a good conversation about the importance of definitions and how sometimes not everyone agrees on a particular definition. After the query, we had JS, RZ, and JR2 presented 2.74, 2.76, and 2.80, respectively. In the last couple minutes of class, I discussed (but did not prove) 2.89, which is the Principle of Mathematical Induction (PMI).

## Week 3

**Monday, September 9:**We cruised through the beginning of the Chapter 3 material today. In the interest of time, we skipped the Daily Query. With the exception of TJ coming to the board, I acted as scribe as students volunteered their answers. Here is what transpired: WW (3.2), MH (3.3), SA (3.4), RZ (3.5), NA (3.6), TJ (3.8). As a class, we discussed 3.9, 3.11, 3.12, and 3.13.**Wednesday, September 11:**Well, we didn’t get through everything I was hoping to, but we got close. We had one quick Daily Query. TL told us about the 4 rules that make up our intuitive definition of a group. In order to get through as many problems as we could, we had many people at the board at once writing down their proposed solutions to the exercises. After people were done writing, we talked about each one as a class. Here is what transpired: KS (3.15), TL (3.16), SB (3.17), SA (3.18), SM (3.19), PT (3.20), DJ (3.21), RG (4.1), NA (4.2), and RZ (4.3). DT and TJ volunteered to present 4.4 and 4.5, respectively, but we didn’t to these. We’ll sort these out next time. We also need to do 3.14.**Friday, September 13:**We probably tried to cover too much today as we were wildly pressed for time at the end. We skipped the Daily Query and jumped right into the outstanding problems from last time. After DT and TJ presented 4.4 and 4.5, respectively, we started the new stuff. I discussed 4.6 and then we had DJ, SM, SF, and JS present 4.7, 4.8, 4.9, 4.10, and 4.11, respectively. Along the way, we discussed lots of various items. We never really got to discuss SF and JS’s solution, but I gave them credit nonetheless. We’ll take time to hash out 4.10 and 4.11 in detail on Monday.

## Week 4

**Monday, September 16:**Other than thinking we had 20 minutes of class left when class was over, I think things went well. We started by reviewing the Cayley diagrams for $S_2, R_3, R_4, D_3, D_4,$ and $\mathrm{Spin}_{1\times 2}$. This took longer than I thought it would, but I feel that it was well worth it. We drew two different Cayley diagrams for $D_3$ on the board. One of these JR did, which took care of 4.13. Next, we had SA, RZ, WW, and KS discuss 4.12, 4.14, 4.15, and 4.16, respectively. We didn’t quite finish 4.16, so we’ll have to start there next time. We outright skipped 5.1, but that’s okay.**Wednesday, September 18:**I spent the first 10 minutes of class discussing the use of LaTeX with writeLaTeX.com. After that we discussed the remaining parts of 4.16. With the time we had left, JS presented both 5.2 and 5.3, MH discussed the key ideas for 5.5, and then DJ presented 5.6 and 5.7. At the request of a few students, I wrote down a formal proof for 5.5. Exercises 5.8, 5.10, 5.11, 5.12, and 5.13 are still open. We’ll discuss these on Friday.**Friday, September 20:**We had a pretty good day. MH, NA, KS, SB, JR, and SM presented 5.8, 5.10, 5.11, 5.12, 5.17, and 5.18, respectively. As a class we discussed 5.14. We skipped 5.13, but I’ll discuss that one on Monday.

## Week 5

**Monday, September 23:**Class began with a brief discussion of our upcoming exam. We hadn’t done a Query in a while, so I decided to squeeze one of those in. NA was the luck one chosen to discuss subgroups. After that we discussed 5.13 and 5.19 as a class. Next DJ, TL, and JR2 presented 5.20, 5.22-5.23, and 5.24, respectively. We never made it to 5.25, so I agreed to post a solution on the course forum in Canvas.**Wednesday, September 25:**Another big day! I wasn’t sure that we’d get through everything, but we got darn close. Here’s what transpired: WW (6.8), RZ (6.9), PT (6.10), JR (6.11), SB (6.13), TJ (6.14), TL (6.16), SM (6.17), and JR2 (6.20). In addition, we discussed 6.21 and 6.22 as a class. We didn’t quite get to 6.23, so I’ll post some solutions on Canvas.**Friday, September 27:**The students took the in-class portion of Exam 1.

## Week 6

**Monday, September 30:**Since the students are working on the take-home portion of Exam 1, I lectured over content from Chapter 6.**Wednesday, October 2:**I continued lecturing over Chapter 6.**Friday, October 4:**Our first day of student presentations following the first exam. I was grateful to see that everyone easily got back into the swing of things. We started with a Daily Query, which we haven’t done in a while. JS addressed the question, “When are two groups isomorphic?” SB tackled, “What does it take to show that two groups are not isomorphic?” SM was asked the following question. “Suppose $|G|=|H|$ and $G$ and $H$ both have generating sets of size two. Does this imply that $G\cong H$?” After the Query, JS, MH, TJ, TL, WW, SB, and JR discussed/presented 6.36, 6.37, 6.38, 6.39, 6.40, 6.41, and 6.42, respectively. With the time we had left, we discussed 6.43-6.45 as a class.

## Week 7

**Monday, October 7:**We began with a quick review of the topics we’ve been discussing. After that we discussed the problem from Exam 1 that asked us to draw a Cayley diagram for $D_4$ that “reveals” $V_4$ as a subgroup. Next, SA tacked 6.47 and then we discussed 6.48, 6.49, 6.50, and 6.53 as a whole class. We skipped 6.52, but we’ll come back to it next time.**Wednesday, October 9:**I finally gave back the take-home portion of Exam 1! However, I’m still suffering a tremendous amount of guilt from the fact that I still have the in-class portion. We started off by discussing the notion of a subgroup generated by a subset. Then we launched into student presentations. TL put 6.52 to rest, SB addressed all four parts of 6.54, TJ drew a wonderful picture for 6.55, RG discussed 6.56, and SA provided a counterexample for 6.58. Exercise 6.56 seemed to be controversial, but I’m hoping that the things I said in class cleared up any confusion. We didn’t make it to 6.59 and 6.61. We’ll start with those ones next time.**Friday, October 11:**Class began with me doing the proof for 6.59, which was outstanding from last time. SB tackled 6.61, which was our second outstanding problem from last time. After that we had WW, MH, RZ, JR1, and DJ discuss 6.62, 6.63, 6.64, 6.65, and 6.66, respectively.

## Week 8

**Monday, October 14:**Busy day today. We didn’t quite get everything done, but we got close. MH, SF, JR2, SB, DT, and DJ put 6.67, 6.68, 6.69, 6.71, 6.72, and 6.75, respectively, up on the board and then we discussed them. As we went, we discussed 6.70 and 6.74 as a class. 6.76 and 6.77 are outstanding and we’ll do these next time.**Wednesday, October 16:**Things seemed a bit chaotic today, but I think it went well. The concepts we discussed are generally difficult for students and most people seemed to be on point. We had SM, WW, PT, MH, and DJ present 6.76, 6.77, 6.78, 6.79, and 6.80, respectively. We never got to 6.81 and 6.82, but TL and TJ will tackle these next time.**Friday, October 18:**Dr. Swift covered for me while I was out of town. TL, TJ, and SM presented 6.81, 6.82, and 6.83, respectively. Rumor has it, most of the class time was spent on 6.83.

## Week 9

**Monday, October 21:**Class began with a discussion of 6.83 and then I wrote down a formal proof. After that we had RZ, SB, DT, SA/SF, and DJ discuss 7.1, 7.2, 7.4, 7.5, and 7.6, respectively. As a class, we quickly discussed 7.7.**Wednesday, October 23:**I really enjoyed today’s class. We started off with SF presenting a proof of 7.8. Next, KS tackled 7.9, which generated lots of discussion. After that we blitzed through 7.10, 7.12, 7.13, and 7.14, which SB, MH, TJ, and DJ presented, respectively. We didn’t get to 7.15 and 7.16. RZ will start off with 7.15 on Friday.**Friday, October 25:**We first had RZ crush both 7.15 and 7.16. Next, we had PT, RG, DJ, and SA present 7.18, 7.19, 7.20, and 7.21, respectively. 7.22 and 7.23 are outstanding, but we may skip covering them.

## Week 10

**Monday, October 28:**The students took the in-class portion of Exam 2.**Wednesday, October 30:**Dana lectured on content related to cyclic groups.**Friday, November 1:**We wrapped up our discussion of cyclic groups and also discussed the dihedral groups.

## Week 11

**Monday, November 4:**A whirlwind tour of permutations and the symmetric group. We didn’t cover all of the problems, but we did have JS, TJ, SA, SB, and WW present 7.26, 7.29, 7.30, 7.35, and parts of 7.36, respectively.**Wednesday, November 6:**Today was…interesting. Weird energy in the room. We postponed 7.39 and will do it next time. We started off with me writing down an example related to 7.41. Next up, PT crushed 7.42 and 7.43. Then we muddled through 7.44, 7.45, and 7.46, which were presented by NA, RZ, and SF, respectively. With me as scribe, we jotted down ideas for 7.49 and 7.50, which are related to Cayley’s Theorem.**Friday, November 8:**We began class with TL presenting 7.39, which was outstanding from last time. Next up, we had JR discuss both 7.61 and 7.62. After discussing some of the problems from the notes, we finished by having PT and SB present 7.67 and 7.68, respectively.

## Week 12

**Monday, November 11:**Veteran’s Day, no class.**Wednesday, November 13:**Busy, busy day. We started with discussing an overview of most of Chapter 7 and along the way tackled some of the exercises and theorems in the notes. Then we had MH present 8.8(b) and DJ present 8.12. There is a left over detail from 8.12, which I promised to address in the course forum. Also, I agreed to post something about 8.19 in the forum.**Friday, November 15:**We blitzed through a discussion of nearly all the relevant problems. Afterwards, we had SA, MH, and TL present 8.27, 8.29, and 8.33, respectively.

## Week 13

**Monday, November 18:**After a lengthy discussion of most of the relevant exercises and theorems, RZ discussed 9.18. DJ got started with 9.10, but we ran out of time. He will start over on Wednesday.**Wednesday, November 20:**I spent half the class session discussing the big picture of quotient groups. After that, we had DJ work on 9.10, which he had started last time. After working on it for a bit, he passed the marker to PT. With some help, we wrote down the sketch of a proof for one direction of 9.10. Next, we had DT and KS working on the two directions of 9.29, respectively. However, what was written down needed some work. I’ll post something about this one and a couple others on Canvas.**Friday, November 22:**The students took the in-class portion of Exam 3.

## Week 14

**Monday, November 25:**I spent most of the class session lecturing over group homomorphisms.**Wednesday, November 27:**After a short “heart-to-heart” conversation, I lectured over the First Isomorphism Theorem. I’ll finish up the proof next week.**Friday, November 29:**Thanksgiving Break! No class.

## Week 15

**Monday, December 2:**We finished up our formal exploration of groups and started discussing rings.**Wednesday, December 4:**We continued our tour of rings.**Friday, December 6:**More lecturing, more rings. The main topic was ideals.

## Week 16

**Monday, December 9:**Another lightning fast lecture about rings. This time we discussed maximal ideals.**Wednesday, December 11:**Last day of classes! I lectured about prime ideals and the First Isomorphism Theorem for Rings. After that we discussed the problems from the last homework assignment. We had quick bonus presentations by MH and TL.