Math 120: Groups and Rings

Fall 2014

Tuesdays and Thursdays 12:50-2:05 in 380-W

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. This is also a Writing in the Major class. Most students interested in this material will find Math 109 (offered in spring quarter) more appropriate.

Professor: Ravi Vakil, 383-Q,

Course assistant: Francois Greer, 381-A,

Office hours: Tuesday 8-10 am and 11 am-noon (Francois Greer), Wednesday 9-10:30 am (Ravi Vakil), Thursday 8-10 am and 11 am-noon (Francois Greer), Friday 10:30 am - noon (Ravi Vakil). (In the week of the final exam, these office hours are replaced by 12 hours of bonus office hours shown below.)

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.

Email lists and study groups. 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 dates: Thursday at 3 pm outside my office. 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.

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. Deadlines to be aware of: Fri. Oct. 31, Friday Nov. 7, Fri. Nov. 21.


  • Tues. Sept. 23: 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. You defined group homomorphism and isomorphism.
  • Thurs. Sept. 25: Examples of groups, including symmetric groups (even infinite ones), matrix groups (linear algebra and group theory), field (e.g. integers mod p). Subgroups. Group actions.
  • Sat. Sept. 27: short online assignment 0 due.
  • Sun. Sept. 28: 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.
  • Tues. Sept. 30: Short online assignment 1 due. Isomorphisms. (Sub)groups geneated by a set. Cyclic (sub)groups, finitely generated subgroups. Subgroups of Z/n? Group actions. Examples, including group action on g. Center, centralizer, normalizer, stabilizer.
  • Thurs. Oct. 2: stabilizer, kernel, faithful. The information of a group action by G on a set A is the same as maps from G to S_A. Cosets of a subgroup. They "tile" the group. Lagrange's Theorem (if H is a subgroup of a finite group G, then |H| (the size=order of H) is a factor of |G|). The order of every element of G divides |G|. Fermat's Little Theorem, and Euler's fancier version of it. In trying to figure out quotients of groups by subgroups, you invented normal subgroups, and quotient groups.
  • Sun. Oct. 5: Read 3.1, 3.2, and 3.3 by now. Short online assignment 2 due.
  • Tues. Oct. 7: Quotient sets and quotient groups. Propositions 13 and 14, and Corollary 15. The first and third isomorphism theorems.
  • Thurs. Oct. 9: The isomorphism theorems. Composition series and simple groups. Introduction to the alternating group. Problem set 1 due.
  • Sun. Oct. 12: Read 3.4, 3.5, 4.1, and 4.2 by now. Short online assignment 3 due.
  • Tues. Oct. 14: WIM topic: impossibility of doubling the cube and trisecting a general angle.
  • Thurs. Oct. 16: Group actions (kernel, stabilizer, faithful, transitive, orbit). Bijection between the cosets of Stab(a) in G, and the orbit of a. Cayley's Theorem. If p is the smallest prime dividing the order of a group G, then any index p subgroup of G is normal. Problem set 2 due.
  • Sun. Oct. 19: Read 4.3, 4.4, 4.5, and 4.6 by now. Short online assignment 4 due.
  • Tues. Oct. 21: class equation, automorphisms of groups, conjugation in the symmetric group, inner automorphisms, toward the simplicity of the alternating group A_n.
  • Thurs. Oct. 23: More on automorphisms of groups. Simplicity of A_n. Problem set 3 due.
  • Sun. Oct. 26: Read 5.1 and 5.2 by now (and reread 4.5). Short online assignment 5 due.
  • Tues. Oct. 28: Statement of Sylow's Theorem. Applications, including classification of simple groups of size less than 60.
  • Thurs. Oct. 30: Proof of Sylow's Theorem. Problem set 4 due.
  • Fri. Oct. 31: WIM drafts handed in (by noon) to Francois Greer's mailbox by anyone who wants feedback in advance of next week's deadline.
  • Tues. Nov. 4: Francois Greer teaches: direct products; classification of finite abelian groups; the fact that the units mod p are cyclic.
  • Wed. Nov. 5: No office hours (Ravi Vakil away).
  • Thurs. Nov. 6: Prof. Sound teaches: identification of direct products; semidirect products.
  • Fri. Nov. 7: WIM drafts handed in (by noon) to Francois Greer's mailbox. Drafts may not be handwritten.
  • Tues. Nov. 11: midterm (in class). Here is a practice midterm. It will cover up until the statement and use of the Sylow theorems (and will include simplicity of the alternating group, which was done earlier).
  • Thurs. Nov. 13: introduction to rings. Ideals.
  • Sun. Nov. 16: read 5.4, 5.5, 7.1-7.4 by now.
  • Tues. Nov. 18: properties of ideals. Euclidean domains are principal ideal domains.
  • Thurs. Nov. 20: problem set 5 due.
  • Fri. Nov. 21: WIM final papers handed in (by noon) to Francois Greer's mailbox.
  • Sun. Nov. 30: read 7.5-8.3 by now.
  • Tues. Dec. 2: unique factorization, and Fermat's two-square theorem.
  • Thurs. Dec. 4: group theory and the Rubik's cube; review of class. Problem set 6 due.
  • Thurs. Dec. 11: final exam, 3:30-6:30 pm in 380-C. Here is a practice final (with the typo in problem 7 fixed). This week: intead of the usual office hours, we will have bonus office hours: Sunday Dec. 7, 6-9 pm (Francois Greer); Tuesday Dec. 9, 2-4 pm (Ravi Vakil) and 6-9 pm (Francois Greer); Wednesday Dec. 10, 10:45 am - 12:45 pm (Ravi Vakil); Thursday Dec. 11, 10:15 am - 12:15 pm (Ravi Vakil).

    Miscellaneous: Josh Meisel (who took 120 in 2009) found this link about a Futurama episode whose plot involved group theory.

    To my home page.