Here are solutions to the final exam:

Final Exam Solutions

I've posted final exam scores and course grades to BB Learn, along with a scan of the final distribution of scores and grades. You can come by my office after January 1 to look at your exam. I am supposed to keep them for one year - I'd be happy to return them after that.

Here are links to the two recent papers I mentioned a few weeks ago, one concerning (infinite) groups with few conjugacy classes, and the other concerning very large free groups:

Shahryari, Algebraically closed groups and embeddings

Bruno, Pathological and highly transitive representations of free groups

I will also post my old notes on character theory and the proof of Burnside's Theorem once I get them scanned, in a few days. If you're looking for more mathematics to read over break, one possibility is Chapters 11 and 12 of Rotman - you should be able to understand it, and the subject matter is pretty fun and interesting. Another possibility is the book by M.A. Armstrong called "Groups and Symmetry."

I uploaded Andrew's nicely TeX'ed solution to Exercise 5.46, concerning the orders of simple groups of size less han 60 - click onthe number below to see it. Thanks to Andrew for the nice table.

Here are my notes about maximal ideals and maximal subgroups:

Maximal ideals and maximal subgroups

If you have any comments, corrections, good examples, or more theorems, let me know and I'll edit the file.

Students have observed some generalizations, and some interesting questions have arisen, in connections with some of our homework problems and exercises. Instead of discussing them in class, I have written brief descriptions - click to see.

centralizers and normalizers of cyclic subgroups of the symmetric group

groups with few conjugacy classes

Here are links to pdf's of some of the figures related to the icosahedron, and to the article about Klein's solution to the quintic (with a slightly simpler argument), as in Lectures on the Icosahedron (1884).

Figures

Oliver Nash's article

I wrote up the solution to Problem 2.50 of the text, even though it wasn't assigned. It is a fundamental result, and is relevant for Problem 6 on Exam 1, so I thought it would instructive perhaps to have the argument available:

Solution to Problem 2.50

It was pointed out to me that Problem 2.11 of the text is in error. The hypothesis should be that X and Y are subgroups of G. (XY need not be a subgroup.) I've posted some exercise solutions - click on the number to see the solution. E-mail me with requests for exercise solutions, if you have them. I made a small error in the solution to HW #2.4; the solutions below have now been corrected.

My story about Malcev's death may be apocryphal (in particular, not true). Here is a more anodyne version:

A brief biography of Anatoly Malcev

A comment or two on some of the Chapter 1 exercises: the problems about cancellation semigroups are motivated by applications to ring theory. Namely, if (R, + , *) is a ring, then (R, *) is a semigroup, and it is a cancellation semigroup iff R is a (commutative) integral domain, i.e., R has no zero-divisors. Problem 1.31 is a generalization of the familiar fact that a finite integral domain is a field. Problem 1.32 is a generalization of the familiar fact that an integral domain can be embedded in a field, its field of fractions.

Syllabus (includes tentative exam dates)

Exams

• Exam 1
  • Solutions
• Exam 2
  • Solutions
• Exam 3
  • Solutions

Problem Sets

• HW #1, due 9/6
  • Solutions
• HW #2, due 9/13
  • Solutions
• HW #3, due 9/20
  • Solutions
• HW #4, due 10/11
  • Solutions
• HW #5, due 10/18
  • Hints: 1
  • Solutions

• HW #6, due 11/15
  • Solutions
• HW #7, due 11/22
  • Solutions

• HW #8, due 12/9
  • Solutions

Exercises (from the text The Theory of Groups: An Introduction, by Joseph Rotman, 2nd ed.)

Ch.1 / 2, 4, 15, 18, 19, 20, 21, 25.

The exercises from Chapter 1 listed above are due Wednesday 9/4.

Ch.1 / 26, 30, 32, 33, 35, 36.

The exercises from Chapter 1 listed above are due Wednesday 9/11.

Ch. 2 / 4, 11, 15, 17, 18, 19, 20, 21.

The exercises from Chapter 2 listed above are due Wednesday 9/18.

Ch. 2 / 27, 29, 30, 33, 39, 40, 41, 43, 46, 52.

The exercises from Chapter 2 listed above are due Wednesday 9/25.

Ch. 2 / 54, 55, 56, 57, 58, 59, 60, 61.

The exercises from Chapter 2 listed above are due Wednesday 10/9.

Ch. 3 / 10, 13, 14, 15, 18, 19, 23, 24.

The exercises from Chapter 3 listed above are due Wednesday 10/16.

Ch. 3 / 30, 36, 37, 39, 41, 43, 51.

Ch. 4 / 4, 5, 6, 7, 8, 10, 12, 13.

The 15 exercises from Chapter 3 and 4 listed above are due Wednesday 10/23.

Ch. 4 / 15, 71.

Ch. 5 / 2, 11, 12, 13, 15, 16, 17, 20.

The exercises listed above are due Friday 11/8.

Ch. 5 / 24, 28, 37, 44, 46.

Ch. 6 / 6 (note the italicized remark preceding the exercise), 11, 12, 14, 24, 28, 38.

The exercises listed above are due Monday 12/9.