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| Quick links to: | Announcements | General information | Assignments | References | External resources |
12/09/05 Final Exam posted The final exam is now available! Solutions are due (either directly to me, or to my mailbox on the third floor) by 5:00pm on Friday December 16.
11/01/05 Midterm II posted The second midterm is now available! Solutions are due at the beginning of class on Friday November 11. Those solutions will be scrawled on by me, and returned to you by Wednesday November 16 for rewriting. Your final write-up will then be due Wednesday November 30.
09/28/05 Universal property of quotients I just found the following in Saunders Mac Lane's book: `Categories for the Working Mathematician' (Springer, 1971). At the bottom of page 57:
Again, let N be a normal subgroup of a group G. The usual projection p : G → G/N which sends each g in G to its coset pg = gN in the quotient group G/N is a universal element for that functor H : Grp → Set which assigns to each group G' the set HG' of all those homomorphisms f : G → G' which kill N (have fN = 1). Indeed, every such homomorphism factors as f = f'p, for a unique f': G/N → G'. Now the quotient group is usually described as a group whose elements are cosets. However, once the cosets are used to prove this one "universal" property of p : G → G/N, all other properties of quotient groups - for example, the isomorphism theorems - can be proved with no further mention of cosets [...]. All that is needed is the existence of a universal element p of the functor H. For that matter, even this existence could be proved without using cosets.
09/02/05 Assignment 0 posted I've posted the first list of assignment problems in the Assignments section below. The problems listed should be completed and handed in at the beginning of class on Friday, September 9.
08/24/05 Course web page now online!
Course description. This is the first half of a two-semester graduate abstract algebra course. The first semester will concentrate on group theory, moving on to field theory and Galois theory. (The second semester should cover ring theory, and a little bit of algebraic geometry.) I'll introduce groups and group homomorphisms; subgroups, quotients and products; and group actions. Highlights of the course will include Sylow's theorems, the structure theorem for finitely-generated abelian groups, and the fundamental theorem of Galois theory. I also hope to include some applications of group theory to geometry and topology.
Prerequisites. Students should have taken an undergraduate-level abstract algebra course and a linear algebra course. The course will be essentially self-contained, but previous encounters with some of the ideas involved should render the course material more easily digestible.
Assessment. The final grade will be composed of grades for coursework, two midterm exams, and a final exam, in the following proportions.
Coursework will be assigned every Friday, and completed assignments will be collected at the beginning of class on the following Friday. Late homeworks will not be accepted without prior arrangement. I expect all homeworks to be presented neatly and carefully. Students are encouraged to discuss the homework problems with each other, but writing up should be done individually.
The exam dates are provisionally as follows.
Textbook. The text for the course (both semesters), is `Abstract Algebra', 3rd edition, by David S. Dummit and Richard M. Foote. (See the references section below for full details.) A copy of the 2nd edition should serve just as well as far as content goes, though you may need to look at the 3rd edition for any homework exercises assigned from the book.
Assignments, and solutions to selected assignment problems, will be posted in this section. Paper copies of the assignments are available on request; please let me know if you have any difficulties downloading or viewing the assignments. Problems marked with a * are optional and will not be graded.
| Assignment | Problems | Due date | |
|---|---|---|---|
| Assignment 0 | Section 1.1: 6, 7, 9, 18, 31 Section 1.2: 11, 12 Section 1.3: 2, 9 (generalize this!), 15 Section 1.6: 2, 4, 6 |
Friday September 9 | |
| Assignment 1 |
Section 1.4: 7, 8, 11 Section 1.6: 14 Section 1.7: 1, 8, 11, 16, 21, 23 | Friday September 16 | |
| Assignment 2 |
Section 2.1: 1, 4, 6, 8, 16 Section 2.2: 2, 4, 9, 12, 13 Section 2.3: 3, 9, 15, 16, 21 | Friday September 23 | Selected solutions |
| Assignment 3 |
Section 2.4: 8, 11, 12, 19, 20 Section 2.5: 10, 11 Section 3.1: 7, 12, 14, 18 | Friday September 30 | |
| Assignment 4 |
Section 3.2: 9, 14, 16 Section 3.3: 1, 4, 5, 8 Section 3.4: 3, 8. Please also do exercises 3.4.9 & 3.4.10, or give your own proof of the Jordan-Holder Theorem. | Friday October 7 | |
| Assignment 5 |
Section 3.5: 10, 16, 3*, 5*, 7* Section 4.1: 1, 7, 8, 10, 3*, 4*, 5* Section 4.2: 4, 13, 2*, 7*, 8* | Friday October 14 | |
| Assignment 6 |
Section 4.3: 2*, 8*, 12, 13, 30 Section 4.4: 1*, 5, 8, 13, *18 Section 4.5: 8, 15, 16, 22*, 29*, 30*, 45* | Friday October 28 | Selected solutions |
| Assignment 7 |
Section 4.5: 32, 35, 38, 46 Section 4.6: 1, 2, 6 | Friday November 4 | Selected solutions |
| Assignment *8 |
Section 13.1: *1-8 Section 13.2: *3, *4, *7, *10, *16. | None | |
| Assignment *9 |
Section 13.3: *1-5 Section 13.5: *1, *5, *7, *10. | None |
Here are some external links of varying relevance to the course. Let me know of anything that you think should be added here!