Math 504: Modern Algebra

Professor Monty McGovern; Grader Sid Mathur 
Fall 2016


Instructor:
Monty (or William) McGovern
Office: Padelford C-450
Phone: 206-543-1149
Email: mcgovern@math.washington.edu
Office Hours: drop in, or by appointment
Grader email: sidmath@uw.edu
Grader office hours: Th 1:30-3:30 or by appointment; office ART 336
Lectures:
Monday, Wednesday & Friday, 9:30-10:20 a.m., Padelford Hall C-36
Required Text:

none, but Dummit and Foote's text is on reserve in the Math Library. Lectures will be posted online at http://www.math.washington.edu/504au16.

Prerequisites:
Math 404 or the equivalent. 
Grading:
Your grade will be based on weekly homework, collected on Friday (except for the second week) and counting 40%, and a final exam, counting 60%. The final exam will be comprehensive. If you cannot complete a homework assignment on time, you can always turn it in by 4:00 on the Friday it is due to Sid's mailbox. PLEASE turn in WHATEVER YOU CAN rather than nothing. In the final you may use two letter-sized pages (one sheet front and back of notes in your own handwriting).
Incompletes and Drops:
The grade of Incomplete will be given ONLY if a student has been doing satisfactory work until the end of the quarter and then misses the final exam for a documented illness, religious reason, or family emergency.
What to Expect:
I will begin with group theory, starting with group actions on sets and the Sylow theorems and moving on to the Nielsen-Schreier theory of free groups, very rarely seen in courses at this level. I will then treat module theory over commutative rings, including the structure theory of finitely generated modules over a PID, together with some basic homological algebra. I will then treat semisimple noncommutative Artinian rings and their modules, applying their theory to representations of finite groups. I will post lecture notes throughout the course on my homepage (but NOT the course webpage). I will not use a textbook, but may cite various references from time to time over the term.

             Homework

Due:
Problems:
Sep 30
summarize your background in algebra (groups, rings, fields), on a sheet of paper (not graded)
Oct 5 (WED)
posted in 504au16; Sylow theory for finite groups
Oct 14
posted in 504au16; free products, a free subgroup of PSL(2,Z), free subgroup of F_2 of infinite rank
Oct 21
show that determinant 0 implies singular; show that GL_\infty(R) is isomorphic to the sum of two copies of itself; a nonfree Z-module; finitely generated Z-submodules of Q, tensor product of Z-modules
Oct 28
show that any square matrix is similar to its transpose; show that any two square matrices similar over a larger field are already similar over their basefields; find a basis of exterior powers of a free module; show that a Z-module embeds in an injective one
Nov 4
If R is an integral domain with field of fractions K, then any K-vector space is injective over R; example where quotient of K-modules are not injective; Schur's Lemma, maixmal ideals via Zorn's Lemma, rings R with every left module projective
Nov 10 (THU)
10.5.1,2 and 17.1.1,3 in Dummit and Foote; show the existence of a chain homotopy between projective resolutions of two R-modules and the 0 map between them
Nov 18
Show that all irreducible reps of a finite group G are 1-dimensional if and only if G is abelian; show that the quaternion and dihedral group of order 8 have the same character table; how more generally that the two nonabelian groups of order p^3, p a prime, have the same character table.
Dec 2
Compute the character tables of A_4, S_4; Dummit and Foote, 17.3.3-5


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