Harvard College/GSAS: 7855
Fall 2014-2015
Hiro Tanaka
Location: Science Center Hall A
Meeting Time: M., W., F., at 12
Exam Group: 11
Office Hours for Hiro: Th 12:30 PM - 2 PM and Fr 1:30 - 3:30 PM.
Office Hours for Theo: Sun 3 PM - 4 PM, Kirkland Dining Hall.
Office Hours for Octav: Wed 4 - 5 PM, Math Department Lounge.
Office Hours for Kevin: Sun 8 - 9 PM, Lowell Dinning Hall.

Section for Theo: Th 5 - 6 PM, Science Center 111
Section for Octav: Wed 5 -6 PM, Science Center 309
Section for Kevin: Tu 3 - 4 PM, Science Center 310

This class will be an introduction to groups and rings. Our study will necessarily find connections to vector spaces and their linear transformations, and to bilinear forms and linear representations of finite groups. If time allows, we will try to touch on how these notions are used in cryptography and mathematical/theoretical physics.

Prerequisite: Mathematics 23a, 25a, 121; or 101 with the instructor's permission. Should not be taken in addition to Mathematics 55a. I will assume you are familiar with linear algebra and mathematical proofs.

1. Wed, Sept 3. Introduction to the class. Groups and group homomorphisms.
2. Fri, Sept 5. More examples of groups, subgroups. Group homomorphisms, kernels, images.
3. Mon, Sept 8. Group actions. Statement of Lagrange's Theorem.
4. Wed, Sept 10. Proof of Lagrange's Theorem. Group actions and orbits. Orders of elements.
5. Fri, Sept 12. Free groups.
6. Mon, Sept 15. Equivalence relations: Another view on orbits and free groups.
7. Wed, Sept 17. Finally, free groups are well-defined. Introduction to quotients.
8. Fri, Sept 19. Quotients. Z/nZ.
9. Mon, Sept 22. First Isomorphism Theorem: Images as quotient groups.
10. Wed, Sept 24. First isomorphism theorem and applications; index.
11. Fri, Sept 26. Cycles and cycle notation.
12. Mon, Sept 29. Alternating group A_n. Conjugacy classes in S_n.
13. Wed, Oct 1. Elliptic Curves, finite generation, Mordell's Theorem.
14. Fri, Oct 3. The fundamental group. End of examples week.
15. Mon, Oct 6. Simple groups, short exact sequences. A_4 is not simple.
16. Wed, Oct 8. Semidirect products.
17. Fri, Oct 10. More on semidirect products.
18. Wed, Oct 15. Midterm 1. Here are the solutions.
19. Fri, Oct 17. Stabilizers and counting formulas.
20. Mon, Oct 20. More counting, First Sylow Theorem.
21. Wed, Oct 22. Proof of first Sylow Theorem.
22. Fri, Oct 24. Second Sylow Theorem and Third Sylow Theorem. Applications.
23. Mon, Oct 27. Proofs of Sylow Theorems.
24. Wed, Oct 29. Guest lecture: Mike Hopkins on symmetries of icosahedron.
25. Fri, Oct 31. Guest lecture: Emily Riehl on universal properties. Kevin's notes, and What Hiro says you need to remember.
26. Mon, Nov 3. Rings and Z/nZ as a ring.
27. Wed, Nov 5. Commutative rings, ideals, quotients, fields.
28. Fri, Nov 7. Fields, modules, and vector spaces.
29. Mon, Nov 10. Bases, dimension, and rank-kernel.
30. Wed, Nov 12. Matrices, determinants, cofactors, invertibility.
31. Fri, Nov 14. Thinking about ideals. Polynomial rings are PIDs.
32. Mon, Nov 17. Principal ideal domains and unique factorization
33. Wed, Nov 19. Finitely generated modules over PIDs.
34. Fri, Nov 21. Algebraically closed fields and Jordan normal form.
35. Mon, Nov 24. Jordan normal form and Cayley-Hamilton.
36. Mon, Dec 1. Cayley-Hamilton wrap-up.
37. Wed, Dec 3. Final lecture.

2. Week two: Kevin's notes from Sept 09
3. Week three: Theo's notes from Sept 18

1. Homework One. Due Mon, Sept 8. Do Problems 1(b),(c), and (h) from Cluster A.
2. Homework two. Due Mon, Sept 15.
   • For Kevin: Problems 1 and 2.
   • For Octav: Problems 3 and 4.
   • For Theo: Problems 5 and 6(d), (e).
3. Homework three. Due Mon, Sept 22.
   • For Kevin: Problem 1.
   • For Octav: Problem 2.
   • For Theo: Problems 3 and 4.
4. Homework four. Due Mon, Sept 29.
   • For Kevin: Problem 1.
   • For Octav: Problem 2 and 3.
   • For Theo: Problems 4 and 5.
5. Homework five. Due Mon, Oct 6. Note that you do not need to turn in the problems marked with an asterisk (for instance, problem 6).
   • For Kevin: Problems 1 and 2.
   • For Octav: Problems 3 and 4.
   • For Theo: Problems 5 and 6.
   
   + Practice problems for the midterm.
    
6. Homework six. Due Mon, Oct 20.
   • For Kevin: Problem 1.
   • For Octav: Problem 2.
   • For Theo: Problems 3.
7. Homework seven. Due Mon, Oct 27.
   • For Kevin: Problem 1.
   • For Octav: Problems 2 and 3.
   • For Theo: Problem 4.
8. Homework eight. Due Mon, Nov 3. Note that you do not need to turn in Problem 4.
   • For Kevin: Problem 1.
   • For Octav: Problem 2.
   • For Theo: Problems 3 and 4.
9. Homework nine. Due Mon, Nov 10.
   • For Kevin: Problems 1 and 2.
   • For Octav: Problem 3.
   • For Theo: Problem 4.
10. Homework ten. Due Mon, Nov 17. This is longer than usual, so it's been posted early.
    • For Kevin: Problems 1 and 2.
    • For Octav: Problem 3 and 4.
    • For Theo: Problem 5 and 6.
    • For Hiro: Problem 7.
11. Midterm Two. Due Wednesday, Dec 3.
12. Practice problems for the Final. (Posted Dec 10, 2014. These contain the practice problems about Cayley-Hamilton as well.) Here are some rough solutions to some of the problems. There may be some computational typos.

1. Solutions to homework 1 - 4.
2. Solutions to Midterm 1.
3. Solutions to Midterm 2. (Uploaded December 7, 2014.)

There will be no textbook that we closely follow in this class. I will post online hand-written notes after every lecture. Regardless, below are a list of resources you should feel free to consult.

Online sources:

• Judson's free text
• Milne's various notes
• Beachy's various notes
• Conrad's notes on various topics
• Dick Gross's lectures

Print sources:

• Algebra by Michael Artin. This is the standard textbook one would otherwise use for this class. Provides a good collection of examples, exercises, and motivation.
• Abstract Algebra by Dummit and Foote. This is a less expository, more formal textbook.
• Undergraduate Algebra by Serge Lang.

Outside reading

As I mentioned in class, there are modern-day applications of the theory of groups. Here are some sources that talk about these applications, all of which are based on the theory of groups.

• Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. (The basis of "Mapper" is to create certain groups out of the shape of data, and to see at what scale these groups change.)
• From 5 to 13: Redefining the Positions in Basketball Topology-based data analysis identifies 13 basketball positions. Here are the accompanying slides.
• The NSA Back Door to NIST. A short article about the algebra behind some common cryptography standards.

Here is also some reading that puts the history of group theory in context:

• On the work of John Thompson. A speech by Brauer explaining why Thompson was awarded the Fields Medal in 1970.
• The Wikipedia page on Galois.