Math 120, Spring 2011

Akshay Venkatesh, MWF 9--9:50. .

The final will be held Tuesday June 7 at 8:30 am (see spring exam schedule) in room 380D. Here is a practice final.

Note! The statement in 9(b) is false as written. You should prove the given statement under the assumption that I is generated by a polynomial f that does not have repeated factors in its factorization.

Summary Groups, rings and field. More detail: 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.
This is a "Writing in the major" (WIM) class.

Text: The course text will be Algebra by Dummit and Foote. We will cover roughly the first 8 chapters.

Office hours: My office hours will be MWF after class (9:50--10:50). The CA is Jeremy Leach. His office hours: Tuesday 3-5, WF 12:15--2:15.

WIM assignment info: Draft due May 16, final version due May 27.

The WIM Assignment is to write a short (around 5 pages) exposition of the classification theoreom for finite abelian groups. (If you want, you can do another topic of your choice, but if so you should discuss it with me to make sure you know what you're getting into.) You can find a statement of a more general theorem in 5.2 and a proof in 6.1. You'll hand in a draft by May 16 to Jeremy Leach; he will get you back comments, and you should hand in the final version by Friday, May 27. You are very welcome to talk to either Jeremy or me about this before the draft to get even earlier feedback; you are also welcome (and encouraged) to hand in the draft earlier. Details.

Homework 2, due Friday April 15: 1.7.8, 1.7.16, 1.7.19, 1.7.23, 2.1.12, 2.1.14, 2.1.5, 2.2.4, 2.2.5, 2.2.12, 2.3.10, 2.3.16, 2.3.21, 2.3.25. Solutions.

Homework 3, posted Monday April 18, due Monday April 25: 2.4.2, 2.4.7, 2.4.8, 2.5.2, 2.5.4, 3.1.1, 3.1.3, 3.1.14, 3.1.32, 3.1.34, 3.1.42. Bonus question: explain why, if A and B are two rotations in three dimensions, the products AB and BA are rotations by the same angle, although possibly through different axes. Solutions.

Homework 4, posted Sunday April 24, not for assessment because of the midterm, but I recommend you try it anyway: 3.2.5, 3.2.7, 3.2.8, 3.2.11, 3.2.12, 3.3.7, 3.3.8, 3.5.4, 3.5.9, 3.5. 11. Bonus question: If G is a group and H a subgroup of index 5, prove that there is a normal subgroup N of G of index at most 120, contained in H.

Homework 5, due Friday May 6: 3.5.5, 3.5.12, 4.1.1, 4.1.6, 4.1.7, 4.1.8, 4.1.10, 4.2.2, 4.2.4, 4.2.7 4.2.11, 4.2.12. Bonus question: Discuss the solvability of the 15 puzzle.

Homework 6, due Friday May 13: 4.3.3, 4.3.5, 4.3.9, 4.3.11, 4.3.13, 4.3.22, 4.3.30, 4.3.36, 4.4.2, 4.4.18.

Homework 7, due Friday May 20: 4.5.4, 4.5.6, 4.5.15, 4.5.18, 4.5.33, 4.6.1, 4.6.2, 4.6.4.
Bonus question: Let f: G-->G' be a surjective homomorphism with kernel K. We assume G is finite. Let P, Q be Sylow p-subgroups of G. Show that f(P) is a p-Sylow subgroup of G'. Show that f(P)= f(Q) if and only if P and Q are conjugate by an element of K. solutions.

Homework 8, due Friday May 27: 7.1.1, 7.1.5, 7.1.11, 7.1.12, 7.1.13, 7.2.1, 7.3.1, 7.3.2, 7.3.6, 7.3.10, 7.4.8, 7.4.15, 7.6.3. solutions.

"Homework 9": (not for assessment, just some more practice problems related to the material in the last week of course): 8.1.7, 8.2.5, 8.3.3, 8.3.6, 8.3.7, 8.3.8.

Lecture outlines.

1. Week 1. (M) Rotations as the basic example of a group. Definition of group. Definition of homomorphism. (W) More examples of groups: rotations preserving a cube, the dihedral group. (F) More examples of groups: The symmetric group, a bit about the sign homomorphism.
2. Week 2. More examples of groups: matrix groups, quaternion group. Group homomorphisms. Group actions.
3. Week 3. (Professor Vakil): Subgroups, lattices of subgroups, groups generated by subsets. Examples of group actions; stabilizer, kernel, left-multiplication, conjugation, centralizer, normalizer. Quotients (a bit).
4. Week 4. (M) Cosets. The picture of cosets tiling the group. (W) More on cosets, group actions on cosets, normal subgroups, basic example: kernel of homomorphism. (F) Quotient groups, the isomorphism theorems.
5. Week 5. (M) The isomorphism theorems, stress on the first one. Every cyclic group is isomorphic to Z or Z/NZ. (W) The sign homomorphism on S_n. Definition of orbit and stabilizer for group actions.
6. Week 6. (M) The orbit-stabilizer theorem and some applications. (W) Conjugation, conjugacy classes, centralizers. Any p-group has a nontrivial center. (F) More applications of conjugacy: A_5 is simple. Conjugation is an automorphism. Stabilizers of different points in the same orbit are conjugate.
7. week 7. (M) Sylow theorem statements and a proof of the existence assertion, first exhibiting a p-Sylow for S_n and then proving general case from this. (W) Proof of the other assertions about p-Sylow subgroups. (F) Applications of the Sylow theorems: groups of order pq with p and q prime.
8. week 8: odds and ends about groups. Start of rings: basic examples, like Z, polynomial rings, matrix rings. Ring homomorphisms, kernels. Start of Chinese Remainder theorem.
9. Week 9: Completion of the proof of the Chinese remainder theorem. Unique factorization in rings: review for Z. Prime ideals and irreducible elements. Euclidean implies PID implies unique factorization.