An Introduction to the Theory of Groups (Math 437 section 01)

Class times: 10:30AM - 11:45AM, TR (JHSN 224)
Office Hours: 2:30PM - 4:30PM, MWF; or by appointment

Most of the general course information can be found in the course information handout. Student grades are maintained on Blackboard.

Course Notes

1. Abstract Groups
[updated on 08.26.15]

2. Examples
[updated on 09.10.15]

3. Subgroups, Cosets, Quotients, and Morphisms
[updated on 09.24.15]

4. Group Actions
[updated on 12.09.15]

5. Sylow's Theorem
[updated on 11.20.15]

Course Log

Jump to week: 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 and beyond

Week 1

[08.27.15] - Thursday - First day!

We started off with introductions and learned about each other's spirit animal. It turns out that we have a lion, a panther, two wolves (one is a teacher wolf and the other is a baker wolf), a panda bear, and a tortoise. We had a nice discussion about how we got good at the things we're good at (it's all about experience!), and then we talked about what we view as good mathematical skills to have (pic below). Eventually, we even did a little mathematics. Baker Wolf and Panda Bear presented their proofs of Theorem 1.2. The proofs were great and generated good discussion. Thanks everyone! Can't wait for Tuesday!

- The Tortoise

Problems/theorems covered: Th. 1.2 (BW,PB)

Due next time: Read up through 1.8. Prove/answer 1.8(2), 1.14, 1.15, 1.17, 1.18. And, in light of the fact that you will be faced with many hard theorems to prove this semester, meditate on and be prepared to discuss your answers to the following questions.

Question: what is the value of making mistakes in the learning process?
Question: how do we create a safe environment where risk taking is encouraged and productive failure is valued?

Presentation goal for next time: Th. 1.8(2), Th. 1.14, Pr. 1.15, Th. 1.17, Pr. 1.18

Week 2

[09.01.15] - Tuesday

Had two fantastic presentations from Panda Bear and Lion! Both were clear and thoughtful; questions and comments were handled super well. The audience, led by Baker Wolf, had great comments and caught a couple of small errors. (An already great talk is made so much better by a great audience!) Excellent job, everyone!

- Panda Bear demolishing 1.14 -

Problems/theorems covered: Th. 1.14 (PB), Th. 1.17 (L), Pr. 1.18

Due next time: Read up through 2.16. Prove/answer: 2.7, 2.12, 2.13, 2.15, 2.16.

Presentation goal for next time: Th. 1.8(2) (TW), Pr. 2.7, Pr. 2.12, Pr. 2.13, Th. 2.15, Th. 2.16

[09.03.15] - Thursday

Three great presentations today by Teacher Wolf, Lion, and Panther. I've been so impressed with how well everyone has organized their board work. Really, the arguments have been super easy to follow, and it's allowed the audience (led again by Baker Wolf!) to stay engaged and ask excellent questions. Fantastic job!! And many thanks to Panther for the beautiful table of cycle types in $S_4$.

- Teacher Wolf cruising through 1.8 -

- Panther's beautiful table of cycle types in $S_4$ -

Problems/theorems covered: Th. 1.8(2) (TW), Pr. 2.7 (L), Pr. 2.12 (P)

Due next time: Read up through 2.30. Prove/answer: 2.16, 2.22 - 2.26. Note: I want you all to take another go at 2.16

Presentation goal for next time: Pr. 1.15, Pr. 2.13 (BW), Th. 2.15 (PB), Th. 2.16, Th. 2.22, Th. 2.24, Pr. 2.25, Th. 2.26

Week 3

[09.08.15] - Tuesday

We started off with Baker Wolf taking us through 2.13. The presentation was fantastic: well laid out, interspersed with helpful examples, and (my favorite) lots of colors! We then moved to Theorem 2.15 about the connection between order and cycle type in $S_n$ that we started to see in 2.12 and 2.13. Panda Bear took a very nice approach to the problem by reducing it to an easier one and did a great job handeling the many technical details that popped up in the proof. We decided that we need a little extra to finish it off, which we'll see next time.

- Baker Wolf laying out 2.13 -

Problems/theorems covered: Pr. 2.13 (BW), Th. 2.15* (PB)

Due next time: Read up through 2.30. Prove/answer: 2.28, 2.29, 2.30 The notes have been updated! Please download the current version.

Presentation goal for next time: Pr. 1.15, Th. 2.15* (PB), Th. 2.16 (TW), Th. 2.22, Th. 2.24 , Pr. 2.25, Th. 2.26 (L), Th. 2.28, Th. 2.29, Th. 2.30

[09.10.15] - Thursday

We started by wrapping up 2.15; there were a surprising amount of subtleties! Teacher Wolf then crushed 2.16. As in 2.15, the approach was to first reduce the problem to an easier one, and everything was extremely well laid out. Great job! Next, Lion gave us a proof (complete sentences and all!) of Theorem 2.26 that was straight "from the book." Excellent! We ended with Baker Wolf leading us through the proof that $GL_n$ is nonabelian whenever $n\ge 2$. It began with a $2\times 2$ example before moving to the general case. Very well done! ...see for yourself!

Problems/theorems covered: Th. 2.15* (PB), Th. 2.16 (TW), Th. 2.26 (L), Th. 2.29 (BW)

Due next time: Read up through 2.39. Prove/answer: everything (i.e. theorems and problems) up through 2.39. The new stuff is about graphs, and many people find this confusing the first time around. Please start early! Also, no one seemed to feel like they got Th. 2.30, so take another crack at that too. The notes have been updated, again! Please download the current version.

Presentation goal for next time: Pr. 1.15, Th. 2.22, Th. 2.24 , Pr. 2.25, Th. 2.28, Th. 2.30

Week 4

[09.15.15] - Tuesday

Great Day! Started out with another go at 1.15, which asks whether or not $|g_1,\ldots,g_r| = \operatorname{lcm}(|g_1|,\ldots,|g_r|)$ when $g_1,\ldots, g_r$ are commuting elements of a group. (We already proved in 1.14 that the LHS divides the RHS). The sticking point today was, more or less, about the following situation: if $g_1^h \neq 1$ and $g_2^h \neq 1$ (where $g_1$ and $g_2$ commute), is it possible that $g_1^hg_2^h = 1$. We also decided that it may be worthwhile to meditate on this question in the specific case of $S_n$. So here are the questions:

If $g_1^h \neq 1$ and $g_2^h \neq 1$ with $g_1$ and $g_2$ commuting elements of an arbitrary group $G$, is it possible that $g_1^hg_2^h = 1$?
If $g_1^h \neq 1$ and $g_2^h \neq 1$ with $g_1$ and $g_2$ commuting elements of $S_n$, is it possible that $g_1^hg_2^h = 1$?

We followed this up with three presentations on cyclic groups, one time disguised as an automorphism group of a graph. Great presentations, and great conversations. Excellent job today.

Problems/theorems covered: Th. 2.22 (P), Pr. 2.23 (P), Th. 2.24 (TW), Pr. 2.25 (TW), Pr. 2.35 (L)

Due next time: Revisit Th. 2.28 and Th. 2.30. Use the extra time to answer the above questions, make your $\LaTeX$ beautiful, and prepare your best presentations yet for the problems below.

Presentation goal for next time: Pr. 1.15, Th. 2.28, Th. 2.30, Th. 2.34, Pr. 2.36

[09.17.15] - Thursday

Nice day today. Panther got us started on 2.28, and then we finished it off as a group courtesy of the so-called "Master Theorem" and "Lion's Theorem" from linear algebra. Then Panda Bear crushed 2.30 (well, except for that whole there's two solutions to $x^2 = 1$ ☺). We wrapped up with another stellar presentation from Teacher Wolf, this time about $D_4$. (We sure appreciate the pictures and the colors!) Great job all around! Lot's of problems for next time, so get started early!

Problems/theorems covered: Th. 2.28 (P), Th. 2.30 (PB), Pr. 2.36 (BW)

For next time: Read through 3.19. Prove/solve: 3.4, 3.5, 3.9, 3.10, 3.15, 3.16. Here's a hint for Th. 3.9. Let $G$ be a cyclic group, and let $H \le G$. Write $G = \langle g \rangle$ for some $g\in G$. Now, every $h\in H$ can be written as a power of $g$. Let $r$ be the smallest positive integer such that $g^r \in H$. Show that $H = \langle g^r \rangle$.

Presentation goal for next time: Pr. 1.15, Th. 2.34 (maybe), Pr. 3.4, Pr. 3.5, Th. 3.9, Th. 3.15, Pr. 3.16

Week 5

[09.22.15] - Tuesday

Fun day, but small group. The presentations were great, and we ended with a nice discussion about the center of $\operatorname{GL}_n(F)$. Panda Bear got it all started, and Lion suggested we focus on the $2\times 2$ case, which we solved! Hopefully we'll see a presentation about the general case next time.

Problems/theorems covered: Th. 3.9 (PB), Th. 3.10 (L), Pr. 3.16 (PB)

For next time: Read through 3.30. Prove/solve: 3.21, 3.23, 3.25, 3.26, 3.29.

Presentation goal for next time: Th. 3.15, Pr. 3.21, Th. 3.23, Th. 3.26, Th. 3.29

[09.24.15] - Thursday

It was a really good day. We got into it with cosets and, I think, got a whole lot more comfortable with them. Many thanks to the presenters as well as the super attentive audience.

Problems/theorems covered: Th. 3.23 (L), Th. 3.29 (BW), Th. 3.15 - started (PB)

For next time: Read through 3.40. Prove/solve: Th. 3.32, Th. 3.34, Th. 3.35, Th. 3.37 , Th. 3.38. (You can take another go at Th. 3.26 and Th. 3.29 too.)

Presentation goal for next time: Pr. 3.21, Th. 3.26, Th. 3.32, Th. 3.34, Th. 3.35, Th. 3.37 , Th. 3.38

Week 6

[09.29.15] - Tuesday

Great day, and we covered a lot of ground. We started with Th. 3.15 about centralizers of elements, which will be increasingly important down the road. Next was 3.15 about determining when cosets are equal. We skipped proving Lagrange's Theorem today, but someone will have the honor and glory of presenting it next time. Instead, Lion took us through 2 applications of Lagrange's Theorem, and Teacher Wolf got us thining about normal subgroups again. Quotient groups are the next topic, and we can expect some fun/hard theorems on the horizon!

Problems/theorems covered: Th. 3.15 - finished (PB), Th. 3.26 (PB), Th. 3.34 (L), Th. 3.35 (L), Th. 3.37 (TW)

For next time: Read through 3.48. Prove/solve: Th. 3.42, Pr. 3.43, Th. 3.45, Th. 3.48. (Please take another go at Th. 3.32 too.)

Presentation goal for next time: Th. 3.32, Th. 3.38 (BW), Th. 3.42, Pr. 3.43, Th. 3.45, Th. 3.48

[10.01.15] - Thursday

We started getting into quotient groups today. It began with 3.38, which sets up the definition of quotient groups, and Baker Wolf nailed it! For the record, I followed it up with some nonsensical remarks. Ignore me; listen to the Wolf. Always. We then moved on to showing that if $G$ modulo a central subgroup is cyclic, then $G$ is abelian. Panda Bear got it all started, and Lion jumped in for the end game. It was fantastic teamwork. Lagrange's Theorem also went down today in a similar fashion: Lion led it off while Panda Bear clarified a crucial point. And the rest of the team was there too, constantly asking questions and making suggestions. Really it was fantastic! We ended by trying to find a nonabelian group with a normal subgroup for which the normal subgroup and the quotient are cyclic. No one was really sure of their work, but Baker Wolf volunteered to get us started. There was some confusion, which was fantastic because it allowed us to clarify some important points, and in the end, Baker Wolf's guess was right on!! A truly excellent day!!

- Baker Wolf with a gorgeous presentation of Th. 3.38 -

- Teamwork! -

Problems/theorems covered: Th. 3.32 (L-PB), Th. 3.38 (BW), Pr. 3.43 (BW), Th. 3.45 (PB-L)

For next time: Read through 3.55. Prove/solve: Th. 3.50, Pr. 3.52, Th. 3.53, Th. 3.54. (Please take another go at Th. 3.42 and Th. 3.48 too.)

Presentation goal for next time: Th. 3.42, Th. 3.48, Th. 3.50, Th. 3.52, Th. 3.53, Th. 3.54

Week 7

[10.06.15] - Tuesday

We began with 3.42, which highlights the fact that many properties of a group pass to subgroups and quotient groups, e.g. being cyclic. Baker Wolf got us started, and Lion jumped at the end with a key observation. Teamwork again! We then began to investigate a possible converse to Lagrange's Theorem. The general question is

If $G$ is a finite group and $p$ is a prime dividing $|G|$, must $G$ have an element of order $p$?

Lion got us going on 3.50, which says that the answer to the question is YES whenever $G$ is cyclic. We were a little confused as to what the problem was saying, and after a short discussion, Panda Bear knocked it out. Teamwork yet again! And speaking of knocking things out... We decided to end by getting our thoughts about 3.53 and 3.54 on the board, so that they could go down next time. While most of us were talking about 3.53, Panda bear quietly wrote his "ideas" for 3.54 (this theorem states that the answer to the above question is YES whenever $G$ is abelian). When we finally looked at what Panda Bear had done, we were in awe. He quickly (since we were out of time) talked us through it, and showed us where he had gotten stuck at the end. It turned out that the "sticking point" was exactly what 3.53 could deal with, and that was that! The proof of 3.54 was starting right at us! We'll definitely need to talk through it again next time. Great job everyone!

- Panda Bear silently working away -

- The glorious aftermath! -

Problems/theorems covered: Th. 3.42 (BW-L), Th. 3.50 (L-PB), Th. 3.54 (PB!! - official presentation next time)

For next time: Take another go at Th. 3.48, Th. 3.52, Th. 3.53, Th. 3.54. I really want you to understand these. Work together, and make them beautiful! (Of course, you can talk with me too...)

Presentation goal for next time: Th. 3.48, Th. 3.52, Th. 3.53, Th. 3.54

[10.08.15] - Thursday

Very nice day - two fantastic presentations about two fantastic theorems! We also had a lengthy discussion about converses to Lagrange's Theorem as well as the notion of $p$-divisibility. Nice job everyone!

Problems/theorems covered: Th. 3.48 (L), Th. 3.54 (PB)

For next time: Read through 3.72. Prove/Solve: Th. 3.64, Th. 3.66, Th. 3.67, Pr. 3.68, Pr. 3.70, Pr. 3.72

Presentation goal for next time: (some subset of) Th. 3.52, Th. 3.53, Th. 3.64, Th. 3.66, Th. 3.67, Pr. 3.68, Pr. 3.70, Pr. 3.72

Week 8

[10.13.15] - Tuesday

It was a bit of a low energy day, but we still had several very nice presentations. Baker Wolf and Panther started us off with nonexamples and examples of isomorphic groups. Both were right on and to the point. Then Lion delivered an absolutely spotless proof of the fact that the homomorphic image of a cyclic (resp. abelian) group group is cyclic (resp. abelian). While Lion was writing, we started to make a list of the groups of small order that we know. What we came up with is below. We ended by investigating Pr. 3.72 as a group. Nice job everyone.

1	2	3	4	5	6	7	8
trivial group	$C_2$ or $\mathbb{Z}_2$ or $S_2$	$C_3$ or $\mathbb{Z}_3$	$C_4$ or $\mathbb{Z}_4$	$C_5$ or $\mathbb{Z}_5$	$C_6$ or $\mathbb{Z}_6$	$C_7$ or $\mathbb{Z}_7$	$C_8$ or $\mathbb{Z}_8$
			$\mathbb{Z}_2\times \mathbb{Z}_2$		$S_3$		$D_8$
							$Q_8$

Problems/theorems covered: Th. 3.64 (L), Pr. 3.68 (P), Pr. 3.70 (BW), Pr. 3.72 (team effort)

For next time: Read the rest of Section 3 as well as 4.1 and 4.2. Prove/Solve: Pr. 3.73, Th. 3.75, Th. 3.77, Th. 3.78, Pr. 3.80, Pr. 4.2

Presentation goal for next time: (some subset of) Th. 3.52, Th. 3.53, Th. 3.66, Th. 3.67, Pr. 3.73, Th. 3.75, Th. 3.77, Th. 3.78, Pr. 3.80, Pr. 4.2

[10.15.15] - Thursday

Fall Break!!

Week 9

[10.20.15] - Tuesday

Started off with Lion taking us through Pr. 3.73 and got to see some new subgroups of $\operatorname{GL}_2(F)$. Next, Teacher Wolf took down Th. 3.52 about $p$-divisibility, which had been waiting for a proof for a while. It seems like we understand the $p$-divisibility condition better now, but we are still waiting to see how it can be useful. (A proof of Th. 3.53 may do this though.) We ened with Th. 3.75 about kernels and images of homomorphisms. Nice job today!

Problems/theorems covered: Th. 3.52 (TW), Pr. 3.73 (L), Th. 3.75 (PB - started)

For next time: Read up through 4.7. Prove/Solve: Pr. 4.2, Pr. 4.4, Th. 4.5, Th. 4.7

Presentation goal for next time: (some subset of) Th. 3.53, Th. 3.66, Th. 3.67, Th. 3.77, Th. 3.78, Pr. 3.80, Pr. 4.2, Pr. 4.4, Th. 4.5, Th. 4.7

[10.22.15] - Thursday

Very nice day! Started by wrapping up 3.75, and then Panther took us through 3.77 with an excellent, clear presentation. While the speakers were getting ready, we also taked a bit about group actions and transitivity. Hopefully Th. 4.5 will be a little more accessible now. Next, Lion presented the First Isomorphism Theorem. The result is supremely important, and Lion did a great job of walking us through the many details involved. We finished up with Baker Wolf laying out the first two parts of Pr. 4.2 in her usual super clear, well-illustrated style. Looking forward to seeing the rest next time.

Problems/theorems covered: Th. 3.75 (PB - finished), Th. 3.77 (P), Th. 3.78 (L), Pr. 4.2 (BW - started)

For next time: Read up through 4.14. The Notes have been updated. Please download the newest version! Prove/Solve: Th. 4.5, Th. 4.7, Th. 4.8, Th. 4.9, Pr 4.12, Pr. 4.13, Th. 4.14

Presentation goal for next time: (some subset of) Th. 3.53, Pr. 3.80, Pr. 4.4, Th. 4.5, Th. 4.7, Th. 4.8, Th. 4.9, Pr 4.12, Pr. 4.13, Th. 4.14

Week 10

[10.27.15] - Tuesday

Started by continuing our explorations of group actions of $D_6$ with Baker Wolf delivering excellent presentations of 4.2 and 4.4. Next, Teacher Wolf guided us through 4.5, which simplifies what needs to be checked for transitivity, and then Lion took us through the proof of 4.14, showing us that every subgroup $H$ of a group $G$ gives rise to an action of $G$ on $G/H$. Both of these theorems will be useful down the road. We wrapped things up with Teacher Wolf proving that $\operatorname{GL}_n(F)/\operatorname{SL}_n(F) \cong F^\times$ via the determinant map. Great job today!

- Baker Wolf's gorgeous board work! -

Problems/theorems covered: Pr. 3.80 (TW), Pr. 4.2 (BW), Pr. 4.4 (BW), Th. 4.5 (TW), Th. 4.14 (L)

For next time: Read up through 4.19. Prove/Solve: Pr. 4.15, Th. 4.18, Th. 4.19

Presentation goal for next time: (some subset of) Th. 3.53, Th. 4.7, Th. 4.8, Th. 4.9, Pr 4.12, Pr. 4.13, Pr. 4.15, Th. 4.18, Th. 4.19

[10.29.15] - Thursday

Fantastic day! We started by finally taking down Th. 3.53. Panda Bear came in with some good ideas and volunteered to get the problem started. We were treated to an excellent presentation of what turned out to be approximately 97% of the proof. Everyone got involved once Panda Bear got stuck with Lion joining at the board. It culminated in an "I got it!" from Baker Wolf, and once she explained it, we all had it. Awesome! And then there was more. Next, Teacher Wolf took us through a 4.8 - 4.9 double-header with a nice, well-laid-out explanation of the permutation representation associated to a group action. We ended with a brief discussion about simple groups and how to use group actions (and homomorphisms) to create normal subgroups.

Problems/theorems covered: Th. 3.53 (PB et al.), Th. 4.8 (TW), Th. 4.9 (TW)

For next time: Read up through 4.28. Prove/Solve: Th. 4.21, Th. 4.22, Th. 4.23, Th. 4.28. Also take another go at Pr. 4.15, Th. 4.18, Th. 4.19. The Notes have been updated, and Th. 4.22 now has a hint.

Presentation goal for next time: (some subset of) Pr 4.12, Pr. 4.13, Pr. 4.15, Th. 4.18, Th. 4.19, Th. 4.21, Th. 4.23, Th. 4.23, Th. 4.28

Week 11

[11.03.15] - Tuesday

The presentations were a bit rough today, but that is to be expected as we work to get comfortable with group actions. The small errors and bits of confusion generated a lot of productive discussion, which was fantastic! It was really a very nice meeting today! Nice job!

Problems/theorems covered:Pr. 4.15 (BW), Th. 4.18 (PB), Th. 4.23 (TW)

For next time: Read up through 4.35. Prove/Solve: Th. 4.30, Th. 4.33, Pr. 4.35. Also take another go at Th. 4.28.

Presentation goal for next time: (some subset of) Pr 4.12, Pr. 4.13, Th. 4.19, Th. 4.21, Th. 4.22, Th. 4.28, Th. 30, Th. 4.33, Pr. 4.35

[11.05.15] - Thursday

Everyone seemed a lot more comfortable today with the general notion of a group action as well as with action by conjugation. The presentations were very nice today, and, fortunately, there were still a couple of small errors that made for productive discussions. The big punchlines from today were that conjugation is a very natural (and important) action to consider and that the orbits arising from a group action partition the set being acted upon. This latter idea leads into the very important Theorem 4.39, which then specializes to the extremely useful Class Equation for a group. It's all happening now!

Problems/theorems covered:Th. 4.21 (TW), Th. 4.30 (L), Th. 4.33 (PB)

For next time: Read up through 4.42. Prove/Solve: Th. 4.37, Th. 4.39, Th. 4.40, Th. 4.41, Th. 4.42.
Hints: for Th. 4.39, try drawing a picture of $X$ divided up into the orbits of $G$. Then count $X$, orbit by orbit. For Th 4.40, try using Th. 4.39, and remember that all of the numbers in the equation in Th. 4.39 are integers. Lastly, for Th. 4.42, it should be clear that Th. 4.41 may be of help, but there is also an "old" result that is relevant (see Section 3.3).

Presentation goal for next time: (some subset of) Pr 4.12, Th. 4.19, Th. 4.22, Th. 4.28, Pr. 4.35, Th. 4.37, Th. 4.39, Th. 4.40, Th. 4.41, Th. 4.42

Week 12

[11.10.15] - Tuesday

Clementines and doughnuts! Many thanks to the wolves for the yummy treats! We then started with Baker Wolf taking us through Th. 4.22. As usual, Baker Wolf laid out couple of helpful examples and special cases before discussing the general argument. Very nicely done! Next, Panda Bear gave an excellent presentation of Th. 4.28, and I think we all now solid on the difference between normalizing and centralizing. But strangely, Panda Bear didn't prove anything by contradiction ☺. We ended with Teacher Wolf taking us through Th. 4.39. The main idea was well illustrated with a picture, which then led to a discussion about how the equation may be useful in practice. Nice job everyone! Oh and Happy Birthday Lion!

- Baker Wolf! -

Problems/theorems covered:Th. 4.22 (BW), Th. 4.28 (PB), Th. 4.39 (TW)

For next time: Read up through 4.45. Prove/Solve: Th. 4.44, Th. 4.45; you can use 4.43 even though I'm not asking you to prove it. (Also take another go at Th. 4.40-4.42.)

Presentation goal for next time: (some subset of) Pr 4.12, Th. 4.19, Pr. 4.35, Th. 4.37, Th. 4.40, Th. 4.41, Th. 4.42, Th. 4.44, Th. 4.45

[11.12.15] - Thursday

Small group today, but we got good stuff done. Lion crushed the Orbit-Stabilizer Theorem, and Teacher Wolf gave an excellent presentation (complete with a picture) of the fact that groups of order $p^2$ are abelian. Nice job! Also, since everyone had thoughts on 4.44, we took crack at it together. Success!

Problems/theorems covered:Th. 4.37 (L), Th. 4.42 (TW), Th. 4.44 (Team Effort)

For next time: Read up through 5.5. Prove/Solve: Th. 4.46, Pr. 5.3, Pr. 5.4, Th. 5.5. (Also take another go at Th. 4.40 and Th. 4.45.)

Presentation goal for next time: (some subset of) Th. 4.19, Pr. 4.35, Th. 4.40, Th. 4.41, Th. 4.45, Th. 4.46, Pr. 5.3, Pr. 5.4, Th. 5.5

Week 13

[11.17.15] - Tuesday

Started with Panda Bear leading us through Th. 4.46, which was our first application of Cauchy's Theorem. We then transitioned to a couple of problems that got us thinking about Sylow subgroups. Lion laid out 5.4, and then we did 5.3 (and more) as a group. We ended with Lion cleaning up an important theorem about actions of groups of prime-power order. This one is very useful! Nice job everyone! ...and after class I got a surprise visit from a member of last year's seminar - the Boss is back in town!

Problems/theorems covered:Th. 4.40 (L), Th. 4.46 (PB), Pr. 5.3 (Team Effort), Pr. 5.4 (L)

For next time: Read up through 5.7. Prove/Solve: Th. 5.6, Th. 5.7

Presentation goal for next time: (some subset of) Th. 4.19, Pr. 4.35, Th. 4.41, Th. 4.45, Th. 5.5, Th. 5.6, Th. 5.7

[11.19.15] - Thursday

We got started with Baker Wolf leading off a discussion about Th. 5.5 and, and as a result, began working through an example for $S_4$. We took a short break and moved to Pr. 4.35. Here we worked out the conjugacy classes of (elements of) $D_4$, and then went back to investigating conjugacy classes of subgroups in $S_4$. Eventually, we uncovered the picture and, with it, a proof of Th. 5.5. It seems like the struggle paid off, and we all got a better understanding Sylow subgroups and conjugacy. More to come! Have a great break!

Problems/theorems covered:Pr 4.35 (L), Th. 5.5 (BW)

For next time: Read up through 5.11. Prove/Solve: Th. 5.8, Th. 5.11 (of course you can use Th. 5.9 when working on Th. 5.11). And make sure to get a good start on you project. You can use the following template for your final paper (just copy it into your ShareLaTeX account). https://www.sharelatex.com/project/547cb99a9f6ef6366f4494d3

Presentation goal for next time: (some subset of) Th. 4.19, Th. 4.41, Th. 4.45, Th. 5.6, Th. 5.7, Th. 5.8, Th. 5.11

Week 14

[12.01.15] - Tuesday

Started with Panda Bear giving a fantastic presentation of Th. 5.6. Several different topics came together in the proof, which I really enjoyed. Once we digested the content of 5.6, we were able to quickly work through Th. 5.7 together. We ended with a brief discussion of the Sylow Theorems and how they can be useful for solving certain problems. Hopefully we'll see a couple more proofs next time, but make sure to focus on creating a solid outline for your final paper (details below).

Problems/theorems covered:Th. 5.6 (PB), Th. 5.7 (Team Effort)

For next time: Copy the project template to your own ShareLaTeX account using the menu in the upper left. Edit the template to include (a preliminary version of) each of the following, and EMAIL me the resulting pdf file.

Title
Section names (the first will be "Introduction," but you should have at least one more section)
The first 1 or 2 paragraphs of your introduction
All of the definitions that you will need, located in the appropriate sections
The statements of at least 2 of the theorems (or lemmas, propositions, etc.) that you plan to discuss, even if you won't prove them all

Presentation goal for next time: (some subset of) Th. 4.19, Th. 4.41, Th. 4.45, Th. 5.8, Th. 5.11

[12.03.15] - Thursday

We reviewed the Sylow Theorems, as well as some earlier theorems, and started a hunt for simple groups. We began a worksheet about groups of order at most 100, trying to understand which orders may hold a nonabelian simple group. More to come...

Problems/theorems covered:

For next time: Focus on your projects, but it'd be amazing to see solutions to some of the problems below.

Presentation goal for next time: (some subset of) Th. 4.19, Th. 4.41, Th. 4.45, Th. 5.8, Th. 5.8, Th. 5.11, Th. 5.12

Week 15

[12.08.15] - Tuesday

Kept working on the small groups worksheet. We proved several lemmas and ruled out many more orders as possibilites for nonabelian simple groups. It was lots of fun, but the real news was...Cauchy!!! Lion treated us to a very nice presentation of Cauchy's Theorem, claiming one of the gems from the course. Fantastic job!

Problems/theorems covered: Th. 4.45 - Cauchy's Theorem!!! (L)

For next time: Focus on your projects, but it'd be great to see some more of the theorems below.

Presentation goal for next time: (some subset of) Th. 4.19, Th. 4.41, Th. 5.8, Th. 5.8, Th. 5.11, Th. 5.12

[12.10.15] - Thursday

Put more time into the worksheet, and saw new ways to exploit the Sylow Theorems. The magic number was...60! Can't wait for the presentations!

For next time: Final Presentations!!

[12.17.15] - Final Presentation Day!!

A finitely generated, infinite torsion group