Syllabus 250A: Group Theory
Fall 2001

Lectures: MWF 9:00-9:50am, Wellman 109
Discussion section: T 9:00-9:50am, Wellman 111
Instructor: Anne Schilling, Kerr Hall 578, phone: 754-9371, anne@math.ucdavis.edu
Office hours: Monday, Wednesday 10-11, Friday 11-12
Text: D. S. Dummit and R. M. Foote, Abstract Algebra, second edition, published by John Wiley & Sons, 1999.
Problem Sets: There will be weekly homework assignments, handed out on Tuesdays. We will discuss the homework problems in the discussion section the following Tuesday. You are expected to present some of your solutions in the discussion section. Your grade for the discussion section will be based on your participation and your presentation of the homeworks.
Exams: Midterm on November 5; Final exam during final exam period (December 10-15)
Grading: The final grade will be based on: Discussion section 20%, Midterm 30%, Final 50%
Web: http://www.math.ucdavis.edu/~anne/FQ2001/250A.html

Problem sets

Homework 0: Send me an e-mail at anne@math.ucdavis.edu and tell me something about yourself, what kind of math you like, what you expect of the class or anything else, so that I can get to know you all a little bit!

Homework 1: ps or pdf, due October 9
Solutions: ps or pdf

Homework 2: ps or pdf, due October 16
Solutions: ps or pdf

Homework 3: ps or pdf, due October 23
Solutions: ps or pdf

Homework 4: ps or pdf, due October 30
Solutions: ps or pdf

Homework 5: ps or pdf, due November 6
Solutions: ps or pdf

Homework 6: ps or pdf, due November 13
Solutions: ps or pdf

Homework 7: ps or pdf, due November 20
Solutions: ps or pdf

Discussion Section: November 27; we did Problems 9 and 10 in Section 3.4 of Dummit and Foote (pg. 108).

Homework 8: ps or pdf, due December 4
Solutions: ps or pdf

The final exam is on Saturday, December 15, from 8:00 am to 10:00 am in Wellman 109.

Content of the lectures:

The class is based mainly on Parts I and II of the book. Topics include:

Part 1: Group theory
Definition of a group
Groups as symmetries
Examples: cyclic, dihedral, symmetric, matrix groups
Homomorphisms
Subgroups and quotient groups
Cosets
Conjugacy classes
Normal subgroups
Lagrange's theorem
The isomorphism theorems

Actions of groups on sets
Symmetric group and alternating group
Cayley's theorem
Groups of symmetries of plactonic solids
Direct products of groups
Group automorphisms
Sylow's theorem
Applications: classification of groups of small order
The alternating group is simple
Classification of finite abelian groups, finitely-generated abelian groups

Time permitting:
Composition series
Jordan-Hoelder theorem
Nilpotent and solvable groups
Free groups

Part 2: Ring theory
Definition and examples
Ring homomorphisms
Ideals
Chinese Remainder Theorem
anne@math.ucdavis.edu