Abstract Algebra

Professor: Erika L.C. King

Email: eking@hws.edu

Office: Lansing 304

Phone: (315) 781-3355

Home Page

Office Hours: M 1:30pm-3:30pm, W 2:30pm-4pm, Th 11:30am-12:30pm, F 1:30pm-2:30pm, and by appointment

Class Schedule: held TR 1:30-2:55pm in Lansing 301

Course Syllabus

Proof Writing and Presentation Tips

Advice from our textbook's author

Course Grade Scale

**Review Session: Sunday, December 14th 3:30PM-4:30PM in Lansing 301. Bring questions!**

**Office Hours:**

- Sunday, December 14th: 5:00PM-6:00PM
- Monday, December 15th: 12:30PM-1:30PM and 4:30PM-6:00PM
- By appointment

**Final Exam: Tuesday, December 16th 8:30AM-11:30AM in Napier 202. **

**Last set of Reading and Practice Problems:**

- Read pages 177-184 and Cauchy's Theorem on pages 186-187 of Chapter 9 in our text. There are some nice examples here. Remember to keep track of vocabulary.
- Do exercises 1, 2, 7, 10, 12 and 17 from Chapter 9 in the text (pages 191-192).

**Quiz 8 on Tuesday covering everything we discussed in class last week!**

**Reading and Practice Problems for class Tuesday, December 9:**

- Read the article from Notices handed out in class. Remember that I will likely choose a question from the article for the quiz.
- Read pages 137-141 of Chapter 7 in our text. There are some nice examples here. Remember to keep track of vocabulary.
- Do exercises 9, 11, 13, 15, 19, 21, 23 and 25 from Chapter 7 in the text (pages 148-149).

**Reading and Practice Problems for class Thursday, December 11:**

- Read Chapter 7 in our text. There are some nice examples here. Remember to keep track of vocabulary.
- Start reading Chapter 9 in our text. Remember to keep track of vocabulary.
- Do exercises 29, 41, 43 and 45 from Supplementary Exercises for Chapters 5-8 in the text (page 176).
- There may be time for a volunteer to present one of these. I would be interested in someone presenting number 45 (assume beta is in A_9).

**Collected Assignment (Due Friday, December 12 by 3:00PM):**

- Exercises 8, 10, 14, 18, and 26 from Chapter 7 in the text (pages 148-149). Remember to justify your answers!
- There will be no Notebook Problem this week. However, if you choose to do the following bonus problem, you should treat it like one.
- Bonus Problem (10 points): Exercise 22 (note that dealing with the parenthetical remark is much of the proof)
from Chapter 7 (page 149). (Remember:
(1) Turn these in on a separate piece of paper from your other homework, and
(2)
**notebook problems must be done on your own**.). - You may also resubmit any of the Notebook problems that were due November 7, November 21 or December 5. Last chance for all redos! Remember that you must turn in your first draft(s) with any resubmission.

**There will be no quiz this week.**

**Reading and Practice Problems for class Tuesday, December 2:**

- Read Chapter 6 in our text. Remember to keep track of vocabulary.
- Do exercises 19, 23, 25 and 29 from Chapter 6 in the text (page 132).

**Reading and Practice Problems for class Thursday, December 4:**

- Read pages 137-139 of Chapter 7 in our text. There are some nice examples here. Remember to keep track of vocabulary.
- Do exercises 1, 3, 4 and 5 from Chapter 7 in the text (page 148).

**Collected Assignment (Due Friday, December 5 by 3:00PM):**

- Exercises 12 and 14 from Chapter 6 in the text (page 132). Remember to justify your answers!
- Exercises 26, 28 and 30 from Chapter 6 in the text (pages 133-134).
- Notebook Problem: Exercise 10 from Chapter 6 (page 132). (Remember:
(1) Turn these in on a separate piece of paper from your other homework, and
(2)
**notebook problems must be done on your own**.). - You may also resubmit any of the Notebook problems that were due October 31 (last chance for this one!), November 7 or November 21. Remember that you must turn in your first draft(s) with any resubmission.

**Quiz 7 on Tuesday covering everything we discussed in class last week!**

**Note there will be a short collected assignment due Tuesday at the beginning of class!**

**Reading and Practice Problems for class Tuesday, November 25:**

- Read pages 120-127 of Chapter 6 in our text. Remember to keep track of vocabulary.
- Do exercise 3 from Chapter 6 in the text (page 132).

**Collected Assignment (Due Tuesday, November 25 by 1:30PM):**

- Exercises 4 and 6 from Chapter 6 in the text (page 132). Remember to justify your answers!

**Happy Thanksgiving!**

**Since our second exam was Thursday, we will have no quiz this week. **

**Reading and Practice Problems for class Tuesday, November 18:**

- Reread Chapter 5 in our text (pages 94-111). Remember to keep track of vocabulary.
- Do exercises 33, 36-39 and 41 from Chapter 5 in the text (page 114).

**Reading and Practice Problems for class Thursday, November 20:**

- Read pages 120-124 of Chapter 6 in our text. Remember to keep track of vocabulary.
- Do exercises 43, 45, 49, 51 and 53 from Chapter 5 in the text (pages 114-115).
- Do exercises 5 and 7 from Chapter 6 in the text (page 132).

**Collected Assignment (Due Friday, November 21 by 3:00PM):**

- Note that I am not assigning any new problems for Friday, but I will likely give a short assignment due Tuesday.
- Exercises 14 (Explain!), 26, 32, 34 and 44 from Chapter 5 in the text (pages 112-114).
- Notebook Problem: Exercise 46 (page 114) (Remember:
(1) Turn these in on a separate piece of paper from your other homework, and
(2)
**notebook problems must be done on your own**.). - You may also resubmit any of the Notebook problems that were due October 17 (last chance for this one!), October 24 (last chance for this one!), October 31 or November 7. Remember that you must turn in your first draft(s) with any resubmission.

**Our second exam will be in Napier 202 on Thursday, November 13th during our regular class
time. Since we have an exam, we will have no quiz and no
collected assignment due this week. Be sure to catch up on all your reading and practice problems! The
exam will be over Chapters 3-5.**

**Reading and Practice Problems for class Tuesday, November 11:**

- Check out Gallian's true/false quiz for chapters 1-4!
- Read pages 94-102 of Chapter 5 in our text. Remember to keep track of vocabulary.
- Do exercises 9, 11, 19, 21, 25 and 27 from Chapter 5 in the text (pages 113-114).
- Catch up on your practice problems to prepare for Thursday's exam!!!

**Quiz 6 on Tuesday covering everything we discussed in class last week!**

**Reading and Practice Problems for class Tuesday, November 4:**

- Start reading Chapter 5 in our text. Remember to keep track of vocabulary.
- Make sure parts 1-4 on the worksheet from class are complete and you feel comfortable with them. Take a look at parts 5 and 6 also.
- Catch up on your practice problems!!!

**Reading and Practice Problems for class Thursday, November 6:**

- Start reading Chapter 5 in our text. Remember to keep track of vocabulary.
- Do exercises 1, 3, 5 and 17 from Chapter 5 in the text (pages 112-113).

**Collected Assignment (Due Friday, November 7 by 3:00PM):**

- Supplementary Exercises 26 and 28 from Chapter 1-4 section in the text (page 92).
- Exercises 4, 10, 18 (we will discuss the approach to the second part of this in class on Thursday) from Chapter 5 in the text (pages 112-113).
- Notebook Problem: Supplementary Exercise 6 (page 91) (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due October 10 (last chance for this one!), October 17, October 24 or October 31. Remember that you must turn in your first draft(s) with any resubmission.

**Quiz 5 on Tuesday covering everything we discussed in class last week!**

**Reading and Practice Problems for class Tuesday, October 28:**

- Read/reread Chapter 4 in our text. Pay special attention to the proof of the Fundamental Theorem of Cyclic Groups. Remember to keep track of vocabulary.
- Do exercises 9, 13, 29, 39 and 49 from Chapter 4 in the text (pages 82-5).

**Reading and Practice Problems for class Thursday, October 30:**

- Read/reread Chapter 4 in our text. Remember to keep track of vocabulary.
- Do exercises 23, 25, 35, 43 and 47 from Chapter 4 in the text (pages 83-84).

**Collected Assignment (Due Friday, October 31 by 3:00PM):**

- Exercises 10 and 12 from Chapter 4 in the text (page 82).
- Exercises 14, 46, 52 and 54 from Chapter 4 in the text (pages 83-85).
- Notebook Problem: Exercise 20 from Chapter 4 in the text (page 83). (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due October 10, October 17 or October 24. Remember that you must turn in your first draft with any resubmission.

**Quiz 4 on Tuesday covering everything we discussed in class since the exam!**

**Reading and Practice Problems for class Tuesday, October 21:**

- Make a Cayley table for the group of one-to-one functions from a set of three elements onto itself. Compare your table with the table for D3.
- Start reading Chapter 4 in our text. Remember to keep track of vocabulary.
- Do exercises 11, 17 and 21 from Chapter 4 in the text (pages 82-3).

**Reading and Practice Problems for class Thursday, October 23:**

- Keep working on comparing the group of one-to-one functions from a set of three elements onto itself with the group D3.
- Read pages 73-76 of Chapter 4 in our text. Remember to keep track of vocabulary.
- Do exercises 15, 19 and 27 from Chapter 4 in the text (page 83).
- I will be looking for a volunteer to present number 15.

**Collected Assignment (Due Friday, October 24 by 3:00PM):**

- Exercise 52 from Chapter 3 in the text (page 71).
- Exercises 24 and 38 from Chapter 4 in the text (pages 83-84).
- Exercises 18 and 22 from Chapter 4 in the text (page 83). Explain your answers.
- Notebook Problem: Exercise 30 (Hint: Use centralizers) from Chapter 4 in the text (page 84). (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due Friday, September 26 (last chance for this one!), October 10 or October 17. Remember that you must turn in your first draft with any resubmission.

**Have a great break!**

**Since there is no Tuesday this week, we will have no quiz this week. :)**

**Reading and Practice Problems for class Thursday, October 16:**

- Finish the group worksheet from class. We will be beginning class by discussing and having group presentations on different parts of the worksheet.
- Do exercises 1-7 odd from Chapter 4 in the text (page 82).

**Collected Assignment (Due Friday, October 17 by 3:00PM):**

- Exercises 34 and 36 from Chapter 3 in the text (page 69).
- Exercises 38 and 40 from Chapter 3 in the text (page 70). Note that these two go together and that you are asked to make conjectures, but you are NOT asked to prove them. Just give a brief explanation of how you came up with them.
- Notebook Problem: Exercise 24 from Chapter 3 of our text (page 68). (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due Friday, September 19 (last chance for this one!), September 26 or October 10. Remember that you must turn in your first draft with any resubmission.

**Since our first exam was Thursday, we will have no quiz this week. **

**Reading and Practice Problems for class Tuesday, October 7:**

- Continue reading Chapter 3 in our text. Remember to keep track of vocabulary.
- Catch up on Chapter 3 practice problems. Also do exercises 13, 31 and 33 from Chapter 3 in the text (pages 67-69).

**Reading and Practice Problems for class Thursday, October 9:**

- Read Chapter 3 in our text. Remember to keep track of vocabulary.
- Preferably without referring to your text, finish the first side of the groupwork sheet from class, problems 1-5. We will spend time in groups on Thursday, but it is expected that you will just be checking your solutions for 1-5 with your groupmates, not starting from scratch. The majority of class time spent on groupwork should be on the second half of the worksheet.
- Do exercises 15 (some of you were already noticing this result!) and 19 from Chapter 3 in the text (page 68).
- Do Supplementary Exercises 5 and 9 on page 91 (there is a reason I am asking you to do both!). Be sure to look at these two even if it is just to understand what the problems are asking. I am hoping that someone will present them in class.

**Collected Assignment (Due Friday, October 10 by 3:00PM):**

- Complete the problem on this handout.
- Exercises 12 and 18 from Chapter 3 in the text (pages 67-68).
- Exercise 16 from Chapter 3 in the text (page 68).
- Supplementary Exercise 4 on page 90 (note you will need the definition in exercise 3).
- Notebook Problem: Exercise 14 from Chapter 3 of our text (page 68). (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due Friday, September 12 (last chance for this one!), September 19 or September 26. Remember that you must turn in your first draft with any resubmission.

**Our first exam will be in Napier 202 on Thursday, October 2nd during our regular class
time. Since we have an exam, we will have no quiz and no
collected assignment due this week. Be sure to catch up on all your reading and practice problems! The
exam will be over Chapters 0-2 and some of Chapter 3.**

**Reading and Practice Problems for class Tuesday, September 30:**

- Start reading Chapter 3 in our text (pages 59-62). Remember to keep track of vocabulary.
- Do exercises 1, 4, 5, 7, 11, 25 and 27 from Chapter 3 in the text (pages 67-69).
- I will be looking for volunteers to present 4, 7 and 27.

**Reading and Practice Problems for class Thursday, October 2:**

- Prepare the problems on the handout from class.
- Read pages 59-64 of Chapter 3 in our text. Remember to keep track of vocabulary.
- Do exercises 8, 9, 10 and 35 from Chapter 3 in the text (pages 67-69). (These are the ones I mentioned in class.)

**Quiz 3 on Tuesday covering everything we discussed in class last week!**

**Reading and Practice Problems for class Tuesday, September 23:**

- Complete your proofs of the uniqueness of inverses and number 1 on the groupworksheet from class. I will be looking for a volunteer for the latter. We will spend some time in groups finishing the rest of the worksheet, but begin working on it now.
- Read Chapter 2 in our text (pages 42-53). Remember to keep track of vocabulary.
- Do exercises 7, 13, 16, 17, 29 and 33 from Chapter 2 in the text (pages 54-55).

**Reading and Practice Problems for class Thursday, September 25:**

- Complete the groupworksheet from class and be ready to discuss it.
- Re-read Chapter 2 in our text (pages 42-53). Remember to keep track of vocabulary.
- Do keep up with the practice problems! I may choose one or more problems from your practice problems to put directly on your exam.
- Do exercises 11, 21 and 27 from Chapter 2 in the text (pages 54-55).

**Collected Assignment (Due Friday, September 26 by 3:00PM):**

- Exercises 18, 26 and 28 from Chapter 2 in the text (page 55).
- Exercise 22 (explain briefly) from Chapter 2 in the text (page 55).
- Let G be the set of positive real numbers under the operation a*b = the square root of ab. Is this a group? If so, prove it; if not, explain why not.
- Let G be the set of even integers under addition. Is (G, +) a group? If so, prove it; if not, explain why not.
- Notebook Problem: Prove the following: G is abelian if and only if G satisfies middle cancellation (that is, if axb=cxd, then ab=cd). (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.)
- You may also resubmit any of the Notebook problems that were due Friday, September 12 or September 19. Remember that you must turn in your first draft with any resubmission.

**Quiz 2 on Tuesday covering pages 8-22 of our text.**

**Reading and Practice Problems for class Tuesday, September 16:**

- Finish the symmetry worksheet from class.
- Start reading Chapter 1 (beginning on page 31) in our text. Remember to keep track of vocabulary.
- Do exercises 11 from Chapter 0 in the text (page 24).
- Do exercises 4, 5, 7 and 22 from Chapter 1 in the text (pages 37-39).

**Reading and Practice Problems for class Thursday, September 18:**

- Find an example of a binary operation on a set that is common to you. Find an operation on a set that is common to you that is not a binary operation.
- Finish reading Chapter 1 (pages 31-37) and start reading Chapter 2 in our text. Remember to keep track of vocabulary.
- Do exercises 1, 3 and 9 from Chapter 2 in the text (pages 53-54).

**Collected Assignment (Due Friday, September 19 by 3:00PM):**

- Exercise 6 from Chapter 1 in the text (page 37). (Hint: did you do number 7?)
- Prove the following: If a and m are relatively prime, prove that ax is equivalent to b (mod m) will have a solution for any b. (That is, given b there will be an x to make the statement true.)
- Exercises 8, 14 from Chapter 2 in the text (page 54).
- Notebook Problem: Exercise 24 (prove the statement) from Chapter 0 in the text (page 24). (Hint: Use induction.) (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.).
- You may also resubmit any of the Notebook problems that were due Friday, September 12. Remember that you must turn in your first draft with any resubmission.

**Remember to keep your appointments!!!**

**Note that some of the individual
appointments are scheduled during parts of my office hours
so my open office hours on Monday are slightly different this week; they are M: 2:20-3:30. Regular hours resume on
Wednesday.**

**Quiz 1 on Tuesday covering pages 3-7 of our text.**

**Reading and Practice Problems for class Tuesday, September 9:**

- Continue reading Chapter 0 (pages 3-22) in our text. Remember to keep track of vocabulary.
- Do exercises 1, 23 and 49 in the text (pages 23-26).

**Reading and Practice Problems for class Thursday, September 11:**

- Finish any part of the worksheet from class that you did not get to finish in class. We will discuss this at the beginning of Thursday's class.
- Continue reading Chapter 0 (pages 3-22) in our text. Remember to keep track of vocabulary.
- Do exercises 3, 7 and 13 in the text (page 23).

**Collected Assignment (Due Friday, September 12 by 3:00PM):**

- Exercises 8 (Hint: Use the Fundamental Theorem of Arithmetic; there is a second method that does not require FTA as well.), 14 and 16 in the text (pages 23-24).
- Use Exercise 16 to solve the Diophantine Equation 34x+126y=38. Can you find a second solution?
- Exercise 22 in the text (page 24). (Hint: Use induction.)
- Notebook Problems: 10 and 12 on pages 23-24 (Remember: (1) Turn these in on a separate piece of paper from your other homework, and (2) notebook problems must be done on your own.).

**Welcome to Abstract Algebra!!!**

**My calculus students have already signed up for individual appointments. Some of these individual
appointments are scheduled during parts of my office hours
so my open office hours are slightly different this week. My open office hours the rest of this week
are W: 2:50-4:00pm, Th 11:50am-12:30pm and F: 1:30-2:10pm. As usual, if you cannot make these times
and would like to see me, please make an appointment. **

**Collected Assignment (Due Thursday, September 4 at 1:30pm):**

- Write an autobiographical essay as assigned on the syllabus.

**Reading and Practice Problems for class Thursday, September 4:**

- Read the syllabus. Even though we discussed this, you should make sure you are clear on all the policies and expectations.
- As many of you know, I developed a website, Proof Writing and Presentation Tips, for First Steps into Advanced Mathematics. It would valuable for you to use this as a reference when you are preparing your homework and presentations for class.
- Also take a look at this website the author of our textbook made. Some of this uses a lot of the vocabulary I do not expect you know yet, however, he has some good advice for studying mathematics and abstract algebra in general. Read through this site now and refer back to it regularly. Notice that I am not the only one advising you to start assignments early... :-)
- Start reading Chapter 0 (pages 3-5 especially) in our text. Remember to keep track of vocabulary.
- Do exercises 4 and 5 in the text (page 23).
- Note that there will be no additional assignment due on Friday.

Erika L.C. King

Last modified: Friday 12 Dec 16:05:30 EST 2008