Math 120: Groups and Rings
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
Math 120 will be a fast-moving, high-workload class. This is also a
Writing in the Major class.
interested in this material will find Math 109 (offered in spring quarter) more appropriate.
Professor: Ravi Vakil, 383-Q, vakil-at-math.stanford.edu.
Course assistant: Francois Greer, 381-A, fgreer-at-math.stanford.edu.
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!).
Short online assignments 5%
Problem sets 20%
Writing in the Major Assignment 15%
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
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
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.
- Problem set 1 is due on Thursday October 9. Do 12 of the following problems, including at least one from each section. Section 1.1: 1, 23, 26. Section 1.2: 5. Section 1.3: 2 (answers only). Section 1.4: 3, 10. Section 1.6: 6, 18, 20. Section 1.7: 3, 4, 17, 23. Section 2.1: 12, 13. Section 2.2: 3, 10. Section 2.3: 3, 21. Section 2.4: 7, 13.
- Problem set 2 is due on Thursday October 16. Do 12 of the following problems, including at least one from each section in chapter 3. Section 2.3: 9, 23. Section 3.1: 3, 6, 9, 16, 22, 24, 25, 27, 41, 42. Section 3.2: 4, 8, 10, 11, 17. Section 3.3: 1, 3, 4, 7, 9. Section 3.4: 4, 5, 6.
- Problem set 3 is due on Thursday October 23. Do 12 of the following problems, including at least one from each section. Section 3.5: 9, 10, 12. Section 4.1: 1, 4, 9, 10. Section 4.2: 1, 6, 8, 10, 11, 12. Section 4.3: 2a, 4, 5.
- Problem set 4 is due on Thursday October 30. Do 11 of the following, including at least one from each section. Section 4.3: 17, 22, 25, 26, 29, 33. Section 4.4: 1, 2, 3, 8, 16, 18 (worth 2 problems), 19 (worth 2 problems). Section 4.5: 3, 7, 30.
- Problem set 5 is due on Thursday November 20. Do 11 of the following, including at least one from each section. Section 4.6: 1, 6. Section 5.4: 2, 5, 15. Section 5.5: 2, 8. Section 7.1: 14, 23, 26. Section 7.2: 3, 12. Section 7.3: 10, 13, 18, 19, 22, 30, 34.
- Problem set 6 is due on Thursday December 4. Do 10 of the following problems, including at least one from each section excpt 7.5. Section 7.4: 8, 9, 10, 15. Section 7.5: 5 (worth 2). Section 7.6: 1, 5 (worth 2), 7. Section 8.1: 1(d)+2(b) (together worth 1), 3, 6, 7, 12. 8.2: 3, 5. Also, for each of the 6 problem sets, pick one problem you particularly liked, and explain why in a sentence or two.
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.
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