MATH 122: Algebra I: Theory of Groups and Vector Spaces

This course is an introduction to groups and rings, which are foundational concepts in modern mathematics. Along the way, we will also deepen our understanding of linear algebra and the role of structures on vector spaces.

Prerequisites: I will assume you are all familiar with real vector spaces, linear transformations, and matrices. Basic familiarity with mathematical proof is necessary; I recommend that you have taken at least one proof-based class before.

Announcements

Tue, Sep 12. Office hours and sections are being finalized. Follow the link for the schedule.

Tue, Sep 12. See announcement about new room location: Northwest building Room B101. This applies for tomorrow (Wed, Sep 13).

Mon, Sep 11. See announcement about clarifications on optional problems for HW 2.

Sat, Sep 9. Please see announcement about how to submit homeworks.

Friday Sept 1, 10 PM: There was ambiguity in problem 3(b); there, "id_Z" means the identity function from Z to itself. It does NOT mean "identity element of Z." The updated pdf file for the homework reflects this in blue font. 

Friday Sept 1, 10 PM: The notes from the past rendition of my Math 122 class are here.

Friday Sept 1, 10 PM: Please be patient as your CAs and I figure out the best way to split up the grading. It seems easiest to break up the assignments problem by problem, so each CA can grade a different problem without crossing wires. It's okay if your uploaded PDFs/images contain superfluous problems (e.g., if what you upload to "Problem Two" also includes your solutions to Problems One or Three, for instance).

Your first two homework assignments are posted: One assignment is a survey, the other is a problem set. Both are due by Wed, Sep 6.

Finally, feel free to check out Beyond this course , where I have small links to topics/history/ideas that this course has influenced, or has been influenced by.

Lectures, Notes, and Resources

 You can ask in-class questions, or questions in general, here.

Here is a running list of past and upcoming lecture topics (in case you want to read ahead in some of the suggested sources in the syllabus). Chances are I will not cover every topic I want to cover in the next lecture, so topics will start bleeding from one lecture to the next. (Hence only a seven-day forecast at most.) 

1. Wed, Aug 30. First Class. Groups as symmetries. Definitions groups, homomorphisms and isomorphisms. Examples: Z, R, R-0, GLn(R), Sn. Notes by Nat. See Artin Chapter 2.
2. Fri, Sep 1. Basic facts about groups. Group tables. Z/nZ, Abelian-ness, mattress group. [Not covered; will cover later: Basic facts about homomorphisms, subgroups, subgroups of Z. Cyclic groups. Cyclic notation.]  See Artin Chapter 2. Notes by Nat. Addendum from Hiro.
3. Wed, Sep 6. Group actions. Orbits. More examples as needed; perhaps counting formula. (Toward Orbit-stabilizer.) See Artin 6.7. Notes by Nat.
4. Fri, Sep 8. Hiro is out of town; guest lecture. Equivalence relations, equivalence classes. Challenge problems. Here is the hand-out. See Artin 2.7. How was the exercise session? Fill out this survey!
5. Mon, Sep 11. Lagrange's Theorem. Normal subgroups and quotient groups. Artin 2.8 and 2.12 and 6.7. Nat's Notes. [Not covered: All topics listed.]
6. Wed, Sep 13. Normal subgroups and quotient groups.  Artin 1.5. Notes by Nat.  [Topics not covered: Symmetric group. Cyclic notation. Cyclic groups. p-groups.]
7. Fri, Sep 15. Quotient groups. Kernels, images, first isomorphism theorem, universal property of quotient groups. [Topics not covered: Conjugation action. Class equation. Artin 7.2.] Notes by Nat.
8. Mon, Sep 18. First isomorphism theorem, universal property of quotient groups. Cyclic groups. Subgroup generated by a single element, order of elements. Classification fo cyclic groups. [Not covered: Centers. Commutator subgroup. Abelianizations.] (Artin 7.10, 2.12, 2.4) Notes by Nat.
9. Wed, Sep 20. Dihedral group. Generators and relations. (Artin 7.2, exercises for 7.10) [Topics not covered: S_n, Centers, commutator subgroups, abelianizations. Quaternions, dihedral groups. Conjugation action. Class equation.] Notes by Nat.
10. Fri, Sep 22. S_n and A_n. Sign. Subgroups and diagrams of subgroups. Conjugation in S_n. (Artin 7.5, 1.5, 2.2)
11. Mon, Sep 25. Conjugation in S_n. Cayley's Theorem. Abelianizations, commutator subgroups. [Topics not covered: Short exact sequences. Classifying finite groups. Simple groups.] (Artin 2.4, 6.1, exercises for 6.8)
12. Wed, Sep 27. More on quotients; what it means to generate; forget cosets. [Not covered: A_n. Index. Automorphisms of groups. Conjugation action. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups.] (Special Homework Assignment: 20 Questions for Groups.)
13. Fri, Sep 29. Conjugation. Automorphisms of groups. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups. Cauchy's Theorem, index (i.e., applications of counting).[Artin 6.1]
14. Mon, Oct 2. Midterm One given out. Example Week begins: Fundamental groups.
15. Wed, Oct 4. Example week continues: Elliptic curves.
16. Fri, Oct 6. Example week continues: Subgroup of O(3), or symmetries of regular polyhedra in R^3.

• Mon, Oct 9. No class!

17. Wed, Oct 11. Midterm due. Deadline for Classroom-to-table. Subgroups of SO(3).
18. Fri, Oct 13. Equivalence relations. Introduction to rings.
19. Mon, Oct 16. Fields, units. Z/nZ. More rings. Endomorphisms of abelian groups as rings. [not covered: Zero divisors.] (Artin Chapter 11)
20. Wed, Oct 18. Ring homomorphisms. Ideals. Quotient rings. [not covered: Zero divisors. Fields.] (Artin Chapter 11)
21. Fri, Oct 20. Z/nZ, Z/pZ, integral domains, fields, examples. Modules. Linear algebra over arbitrary fields. (Artin Chapters 11 and 14)
22. Mon, Oct 23. Free modules. Submodules. Ideals are submodules. Maps of modules. Kernel, Cokernel, Image, Matrices. Determinants. Generation of modules. Generation of ideals.
23. Wed, Oct 25. Adjugate matrices; invertible linear transformations; Cayley-Hamilton Theorem. GL_n(F) 
24. Fri, Oct 27. Matrices are the same thing as endomorphisms of R^{\oplus n}. [Not covered: Polynomials with coefficients in fields. Integers. PIDs. Irreducibility. Euclidean algorithm for polynomials. Quotients of polynomial rings.]
25. Mon, Oct 30.  Matrices with unit determinants are invertible. [Not covered: Classification of finitely generated modules over PIDs. Classification of abelian groups.]
26. Wed, Nov 1. Finitely generated modules over vector spaces are free. [Not covered: Back to groups. p-Sylow subgroups. Sylow theorems. Applications.]
27. Fri, Nov 3. Cayley-Hamilton Theorem. [Not covered: Short exact sequences. Semidirect products.] Some notes about, including a proof of, the Cayley-Hamilton Theorem.
28. Mon, Nov 6. (If we're not behind: Review session.)
29. Wed, Nov 8. (If we're not behind: Review session.)
30. Fri, Nov 10. In-class midterm.
31. Mon, Nov 13. Sylow theorems.
32. Wed, Nov 15. Applications of Sylow theorems. Cauchy's theorem. 
33. Fri, Nov 17. Semi-direct products.
34. Mon, Nov 20. Statement of classification of finitely generated modules over PIDs.
35. Mon, Nov 27. Application of classification of finitely generated modules over PIDs. Polynomials. Jordan Normal Form.
36. Wed, Nov 29. Final Exam distributed. Lower central series. Why the classification of finite simple groups?
37. Fri, Dec 1. Last Day of classes.

• Thu, Dec 14. Final Exam due.

1. Sometime in the future: What can we do with integers? Rings, subrings of C, modules. (Artin Chapter 11)
2. Direct products and semidirect products. Short exact sequences.
3. Classification of finite groups. Simple groups. Jordan-Holder decomposition.

There are also live-TeXed notes taken by Daniel.