Math 504: Modern Algebra

Professor Monty McGovern; Grader Sid Mathur 
Fall 2016

Monty (or William) McGovern
Office: Padelford C-450
Phone: 206-543-1149
Office Hours: drop in, or by appointment
Grader email:
Grader office hours: Th 1:30-3:30 or by appointment; office ART 336
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

Math 404 or the equivalent. 
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.


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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