Austin Christian

Announcements

• Here's the cheat sheet that will be included with the final exam if (and only if) 85% of enrolled students submit their CIOS for this course. (Let me know if you find errors/typos.) I value your feedback, so even if you're just doing the CIOS for the cheat sheet, I hope you'll help me improve my teaching.
• Our Nov 20 class meeting included two problems not in the textbook. Solutions should appear in your portfolio.
  1. If \(\phi\colon R\to S\) is a homomorphism of rings, describe a ring structure on \(R/\ker\phi\). You need to:
     1. Describe the elements of the set \(R/\ker\phi\).
     2. Define operations on the set.
     3. Prove that these operations satisfy the definition of a ring.
  2. Explain why the structure defined above does not work for \(\mathbb{Q}/\mathbb{Z}\).
• Here's a practice test (PDF) for midterm 2. And here are solutions (PDF).
• On Oct 30, I will present a proof of Theorem 7.21. Even though this won't be presented by a classmate, it still needs to go in your portfolio. You should also prepare a presentation for Problem 7.22, as well as parts (c) and (d) of Theorem 7.29. For those, you may use parts (a) and (b) of Theorem 7.29, as well as any earlier results.
• Here are some sketchy notes about Cayley diagrams for quotient groups, and why the definition of normal subgroup is the way it is.
• We began working on Theorem 4.105 in the October 4 meeting. You are not required to produce a full proof, but should complete the following tasks:
  1. Identify a subgroup of \(S_6\) to which \(D_3\) is isomorphic. To make this helpful for the remaining parts, try to obtain this isomorphism by starting with a Cayley diagram and labeling the vertices with the integers \(1,\ldots,6\).
  2. For an arbitrary group \(G\) of order \(n\), construct a function \(\varphi\colon G\to S_n\) which satisfies \(\varphi(gh)=\varphi(g)\varphi(h)\), for any \(g,h\in G\).
  3. Show that your function \(\varphi\) is injective.
• Here's a practice test (PDF) for midterm 1. The real exam should have roughly the same format, although the precise topics covered may vary. And here are solutions (PDF) to the practice midterm.
• Here's a sample homework solution (PDF), to give you an idea of how thorough you need to be when writing your solutions.
• Here are some important links as we get started:
• Here's a browser-friendly version of the textbook, but theorem numbers below refer to the PDF.
• Be sure to join our class on Discord, and to participate.
• Some portion of your homework grade will depend on good mathematical writing, so you are encouraged to read this essay by Francis Su.
• In case you'd like to use LaTeX for your portfolio, here's (link deprecated) a template Overleaf project you can use to get started. The link takes you to a read-only project; you can clone an editable copy to your own Overleaf Dashboard, or download the files for local editing.