Math 120: Modern Algebra

This class will cover groups, fields, rings, and ideals. More explicitly: Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains.

Math 120 will be a fast-moving, high-workload class. Most students interested in this material will find Math 109 more appropriate. This is also a Writing in the Major class.

Professor: Ravi Vakil, vakil@math, 383-Q, office hours: Monday and Wednesday 4:30-5:30. (Warning: I'll have to leave promptly at 5:30.)

Course assistant: Daniel Murphy, 381-K, dmurphy@math.stanford.edu. Office hours: Tuesday 2-4:30, Wednesday 11-12, Thursday 11:30-2.

Text: Dummit and Foote's Abstract Algebra, third edition (be careful to get the right edition!).

Grading scheme:
Short online assignments 5%
Problem sets 20%
Writing in the Major Assignment 15%
Midterm 20%
Final exam 40%

Short online assignments. There will be weekly short online assignments on Coursework. You needn't answer more than one or two sentences per question. This is intended to be low-stress: you will get full marks for any good-faith answer. This is intended to give us feedback on how the reading has been going. Because I will have to review your feedback in a limited time period on Sunday afternoon, the deadlines will be on Sunday at noon sharp (no submissions will be accepted after that).

Email lists and study groups. I have an email list I use to send urgent emails to the class (e.g. problem set corrections). You can ask to be added or removed from it at any time. Also, there is a list of email addresses and dorm locations of people willing to be contacted to discuss homework or to form study groups. If you are interested in being on the list and seeing the list, just let me know.

Problem sets. There will be weekly homework assignments, posted here. You are encouraged to work together to solve problems. But you must write up your solutions individually, and give credit for ideas that others had. You should give complete proofs. Because the graders will need to process a huge volume of homework in a small amount of time: Please staple your homework, and write your name on each page.

Homework due date: Fridays at 1 pm, in the box outside my office door 383-Q. No lates will be allowed (so the graders can just grade one problem set at a time, and hence have a better chance of getting them back promptly). But to give everyone a chance to get sick, or have busy periods, the lowest problem set will be dropped.

• Problem set 1 is due on Friday October 1. Do 15 of the following problems, including at least one from each section. Section 1.1: 1, 8, 12, 23, 26, 28. Sections 1.2: 5, 10. Section 1.3: 2 (answers only), 8, 10. Section 1.4: 3, 10. Section 1.6: 6, 18, 20. Section 1.7: 3, 4, 17, 23.
• Problem set 2 is due on Friday October 8. Do 8 of the following problems, including at least one from each section. Section 2.1: 3, 6, 12, 13. Section 2.2: 3, 9, 10. Section 2.3: 3, 13, 21. Section 2.4: 7, 13.
• Problem set 3 is due on Friday October 15. Do 13 of the following problems. Section 2.3: 9, 23. Section 2.4: 15, 17. Section 3.1: 3, 6, 9, 14, 16, 22, 24, 25, 27, 41, 42. Section 3.2: 4, 8, 10, 17.
• Problem set 4 is due on Friday October 22. Do 12 of the following problems. Section 3.2: 11. Section 3.3: 1, 3, 4, 7, 9. Section 3.4: 4, 5, 6. Section 3.5: 9, 10, 12. Section 4.1: 1, 4, 9, 10. Section 4.2: 1.
• Problem set 5 is due on Friday October 29. Do 13 of the following problems, including at least one per section. Section 4.2: 6, 8, 10, 11. Section 4.3: 2ac, 4, 5, 17, 22, 25, 26, 29, 33. Section 4.4: 1, 2, 3, 8, 16, 18 (worth 2), 19 (worth 2).
• Problem set 6 is due on Friday November 5. Do 11 of the following problems. Section 4.5: 3, 7, 30, 33, 38, 46. Section 4.6: 1, 3, 5, 6. Section 5.1: 1, 9, 11, 18.
• Bonus: the Mathieu-12 challenge. I have a puzzle in my office called the Topsy Turvy, designed by M. Oskar van Deventer, based on ideas of Igor Kriz, a topologist at the University of Michigan. (You can read the Scientific American article by Kriz and undergraduate Paul Siegel here.) The puzzle has 12 tokens numbered 1 through 12, which start off in order. There are two possible "moves", called Left and Right. Left (L), applied to the starting configuration, permutes them to 11 9 7 5 3 1 2 4 6 8 10 12. Right (R), applied to the starting configuration, permutes them to 2 4 6 8 10 12 11 9 7 5 3 1. The group generated by these permutations is one of the sporadic simple groups, called the "Mathieu 12" group, and is index 5040 in S_12. Your challenge is to tell me how to solve the puzzle. More precisely, write a script (usable on the web so people around the world can use it) that will (a) tell if a configuration is solvable, and (b) explain how to solve it (i.e. give a sequence of L's and R's). Deadline: the end of Thanksgiving break. Reward: the equivalent of 1 problem set, to be added to your problem set grade (not to take you over 100% in the problem sets). You may work in a group, but must then explain how the reward should be split among you.
• Problem set 7 is due on Friday November 12. Do 12 of the following problems, including at least one per section. Section 5.2: 1a+2a (count for one problem), 5, 8, 9, 12. Section 5.4: 2, 4, 5, 15, 20. Section 5.5: 2, 4 (you'll notice that I said something wrong in class!). Section 6.1: 1, 3, 6, 13. Section 7.1: 5.
• Problem set 8 is due on Friday November 19. Do 12 of the following problems, including at least one per section. Section 7.1: 14, 20, 21, 23. Section 7.2: 3, 12. Section 7.3: 10, 18, 22, 25, 29, 34. Section 7.4: 8, 9, 12, 15.
• Problem set 9 is due on Friday December 3. Do 11 of the following problems. Section 7.5: 5. Section 7.6: 1, 7, 11. Section 8.1: 6, 7, 9, 10, 12. Section 8.2: 2, 5. Section 8.3: 1, 4, 5, 6.

Writing in the major assignment. Clear writing is essential to mathematical communication, as you probably realize from reading better and worse mathematical texts. Clarity in writing may be more important in mathematics than in any other science, for a number of reasons. Good exposition is an acquired and important skill. Throughout this class, you'll received feedback on solutions to your problem sets, and you should use this to refine your ability to communicate your ideas clearly and effectively. This writing project will give you an opportunity to focus on your exposition, as opposed to absorbing new mathematical content. This course will emphasize both exposition in communciating mathematics and the structure of proofs. Part of your grade on each assignment and on the exams will be on your exposition of your solutions to problems. Information about the Writing in the Major assignment is here. Dates to be aware of: October 29, November 8, November 15, November 29. Daniel has offered a week extension, so the deadline is now November 8. He also offers to try to return those handed in by October 29 by November 8.

Final Exam: The final exam will take place on Wednesday, December 8 from 12:15 to 3:15 pm in 380-380Y. There will be bonus office hours 2-5 pm on Monday, December 6 (Ravi Vakil) and Tuesday, December 7 (Daniel Murphy). Here is a practice final.

Syllabus.
Mon. Sept. 20: welcome; definition of group; isomorphism of groups. Words you should know: binary operation, associative, commutative, group, abelian, identity, inverse, group operation, cyclic group, symmetric (or permutation) group.
Wed. Sept. 22: More examples of groups; proving some initial properties of groups; the dihedral group; beginning to understand a group by its presentation; fields; groups from linear algebra (the general linear group and the special linear group).
Fri. Sept. 24: short online assignment 0 due.
Sun. Sept. 24: Read up to chapter 2 by now (including the preliminaries chapter). You should know everything in Chapter 1 well except for the generators and relations section of 1.2 (which you should still read). You should try to digest chapter 2, and tell me (on the online assignment) which parts are the hardest to digest. Short online assignment 1 due at noon (ditto for all later ones).
Mon. Sept. 27: what is hard about presentations of a group, symmetric group, fields, group homomorphism, subgroup, lattice, subgroup generated by a subset, cyclic and finitely generated subgroups, classification of symmetric subgroups.
Wed. Sept. 29: group actions, stabilizer, kernel, normalizer, centralizer.
Sun. Oct. 3: read 3.1 and 3.2 by now. Online assignment #2 is due.
Mon. Oct. 4: cosets: how to think about them (clumps / equivalence classes / fibers), "tile" G, form a set, left vs. right, Lagrange's theorem, number theory consequences (Fermat's little theorem and Euler's theorem). Motivation: maps of vector spaces phi: V_1 --> V_2, rank-nullity theorem, Im(phi) = V_1 / Ker(phi). Group homomorphism: kernel subgroup, image subgroup. Normal subgroups and quotient group. Here is a practice midterm. Daniel Murphy bonus office hours 1-2:30 pm.
Tues. Oct. 5: Ravi Vakil's bonus office hour 1-2, then Daniel Murphy's regular office hours 2-4:30 (he'll stay up to an hour later if people are still there and interested at 4:30).
Wed. Oct. 6: Midterm, covering chapters 0 through 2, in class.
Sun. Oct. 10: read 3.3-4.1 by now. Online assignment #3 is due.
Mon. Oct. 11: quotient groups; isomorphism theorems: |HK| = |H||K|/|H cap K|. A central theorem of class: given f: G --> H, we have that G / ker f --> im f is an isomorphism.
Wed. Oct. 13: more isomorphism theorems (the first still rules!), composition series and the Jordan-Holder program, the alternating group.
Sun. Oct. 17: read 4.2-4.4 and skim 4.5 by now. Online assignment #4 is due.
Mon. Oct. 18: group actions, orbits; the bijection between orbits of a in A and the cosets of the stabilizer; Cayley's lame theorem; class equation. Another central theorem: given a group action of G on A, and a in A, we have a natural bijection between the orbit of a and the left cosets of Stab a.
Wed. Oct. 20: automorphisms of groups (and conjugation in S_n); simplicity of A_n (except for one thing).
Sun. Oct. 24: read 4.5-4.6 by now. Online assignment #5 is due.
Mon. Oct. 25: completion of proof of simplicity of A_n for n >= 5; applications of Sylow's theorem.
Wed. Oct. 27: proof of Sylow's theorem; recognition theorem for direct products; toward the classification of finite abelian groups.
Fri. Oct. 29, noon: WIM drafts can be handed in, in Daniel Murphy's mailbox.
Sun. Oct. 31: read 5.1, 5.2, 5.4, 5.5, and 6.1 by now. Much of this you'll already have read for your WIM paper. We won't cover in detail things you are already discussing in your WIM paper.
Mon. Nov. 1: classification of finite(ly generated) abelian groups. Proof of the classification of finite abelian groups (assuming they are products of cyclic groups). Here is a proof of the hard (omitted) part of the classification theorem. I heard this from Peter Diao, a Ph.D. student here. Another approach is discussed here.
Wed. Nov. 3: nilpotent groups and their properties; the lower central and commutator series.
Sun. Nov. 7: read 7.1-4 by now.
Mon. Nov. 8, noon: WIM drafts due in Daniel Murphy's mailbox.
Mon. Nov. 8: rings, homomorphisms, ideals.
Wed. Nov. 10: cancelled due to illness.
Mon. Nov. 15: WIM drafts returned with comments. Special lecture by Prof. Dan Bump on the Rubik's cube. Types of ideals (principal, generated by a subset, finitely generated, sums, products, intersections).
Wed. Nov. 17: isomorphism theorems (esp. 1st and 4th). A series of small but important topics (polynomial rings, fraction fields, chinese remainder theorem, primes and maximals.
Thankgiving break: No class Nov. 22 and Nov. 24.
Sun. Nov. 28: read 7.5-7.6 and 8.1-8.3 by now.
Mon. Nov. 29: Euclidean domains are principal ideal domains, which are unique factorization domains. Examples: Z, k[x], Z[i]. Final WIM papers due in Daniel Murphy's mailbox.
Wed. Dec. 1: Fermat's Two-Square Theorem. You can't double the cube, or square the circle, or trisect with straightedge and compass. WIM papers available for pick-up outside Daniel Murphy's office.
Sun. Dec. 5: last online assignment due.
Mon. Dec. 6: Ravi Vakil's bonus office hours 2-5 pm.
Tues. Dec. 7: Daniel Murphy's bonus office hours 2-5 pm.
Wed. Dec. 8: final exam, 12:15-3:15 (I think in 380-Y).

Miscellaneous: Josh Meisel (who took 120 last fall) gave me this link on a Futurama episode whose plot involved group theory.