Office: Stevenson 313C (Enter through STV 313)
Office Hours: TBA
CATALOG DESCRIPTION: Group theory including the Sylow theorems and
other advanced topics; ring theory.
Prerequisites: MAT 336 or consent of instructor.
TEXT: Abstract Algebra, by D. Dummit and R. Foote, 3rd Edition, Wiley, 2004. (Errata Page in PDF format)
COURSE WEB PAGE: http://www.math.ilstu.edu/gfseeli/mat407/
CLASS NOTES (These class notes will be updated throughout the semester.)
ABOUT THE COURSE: This course is intended as a follow up to an introductory course in group theory. We will review some of the introductory material about groups in Chapters 0 through 2, then go on to study more about the structure of groups with an emphasis on finite groups, including studying group actions and the Sylow Theorems. Further topics in group theory will be covered as well as an introduction to ring theory as time permits.
As a class, we will meet at the scheduled 2:00-3:50 time on Mondays, but this will be supplemented by informal meeting times that will be arranged based on the schedules of the class members and will be set up the first week of class.
GRADING: In this course, you will be graded on weekly homework assignments (50%), a mid-term exam (20%), and a final exam (30%).
EXAMS: The mid-term exam will be a take-home exam that will be distributed on Monday, March 15th and be due the following Monday, March 22nd. The in-class portion of the final exam is scheduled for Wednesday, May 5, from 3:10 to 5:10.
HOMEWORK: Homework will be assigned weekly and collected the following week. Homework problems will be chosed to allow students to probe further into the algebraic concepts we will be covering and allow for further discussion of these topics. Proofs on these homeworks should be written in a clear style and will be graded for both correctness and clarity.
Due Jan 25: Exercises 1 - 9 of notes
Due Feb 1: Exercises 10 - 16 of notes
Due Feb 11: Exercises 17 - 23 of notes
Due Feb 15: Exercises 24, 25 of notes, Dummit and Foote p. 85 #7, p. 95 # 8, p. 96 # 11. Also, find all subgroups (with lattice diagrams) for Z2 x Z5, Z2 x Z6, and D12.
Due Feb 22: pp. 86-88 #4,17,29, p. 96 #12, 18, 19, p. 101 # 3, 4
Due March 4: p. 89 # 36, 38; p. 96 # 16; p.101 # 7
Due March 29: pp. 146-148 # 7, 9 (Assume F3 is Z3), 16, 21, 27, 33
(Hint for #27: Let N, P, and Q be Sylow 3, 5, and 7 subgroups of G. First show NP and NQ are abelian subgroups. Once this is done, you can show N is in the center of G.)
Due April 5: Homework 8.
Due April 15: Homework 9
Due April 22: Homework 10
Algebra on the WEB:
The Development of Group Theory (Article by: J J O'Connor and E F Robertson)
Groups 15 by John Wavrick of UC San Diego. This program allows you to make computations for groups with at most 15 elements.
A discussion of Rubik's Cube Groups can be found at the Dog School of Mathematics.
The Development of Ring Theory (Article by: J J O'Connor and E F Robertson)
First 1000 Primes
The Great Internet Mersenne Prime Search Project (GIMPS)
The Millennium Problems posed by the Clay Mathematics Institute.