MATH 150A (Modern Algebra) - Winter 2023, UC Davis

Lectures:	MWF 1:10-2:00pm, 168 Hoagland (Anne Schilling)
Discussion Sessions:	MAT 150A-A01, CRN 30841, R 6:10-7:00 PM in Hart 1116
Instructor:	Anne Schilling, MSB 3222, anne@math.ucdavis.edu Office hours: Wednesdays 2-3pm; feel free to ask me questions after each class
T.A.:	Mary Claire Simone, MSB 2204 mcsimone@math.ucdavis.edu Office hours: Tuesdays 5-6pm, Wednesdays 3-4pm
Text:	We will mostly follow Michael Artin, Algebra, published by Pearson, second edition, 2011. Another good reference is Dummit and Foote, Abstract Algebra, ISBN 0-471-36857-1.
Pre-requisite:	MAT 67 or (MAT 22A and MAT 108)
Problem Sets:	There will be weekly homework assignments due on Fridays at 5pm on Gradescope. You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you need help with the problems, come to the discussion session and office hours! The best way to learn mathematics is by working with it yourself. No late homeworks will be accepted. A random selection of homework problems will be graded.
Computing:	During class, I will illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using SageMathCloud. Or you can sign up for a Class Account with the math department. Log into cosine.math.ucdavis.edu and type the command `sage` to launch a Sage session in the terminal.
Exams:	There will be one Midterm on Friday February 17 in class. The Final exam will be Friday, March, 24 2023 3:30-5:30pm. There will be no make-up exams!
Grading:	The final grade will be based on: Problem sets 30%, Midterm 30%, Final 40%. Grades will be recorded on Canvas.
Web:	http://www.math.ucdavis.edu/~anne/WQ2023/150A.html
Bed Time Reading:	If you would like some bedtime reading related to math, I can recommend two really good books by Simon Singh: "Fermat's Last Theorem" and "The Code Book". Also, Ed Frenkel's book "Love and Math" is worth a read!

Course description

This course is the first part of a three-quarter introduction to Algebra. Algebra concerns the study of abstract structures such as groups, fields, and rings, that appear in many disguises in mathematics, physics, computer science, cryptography, ... Many symmetries can be described by groups (for example rotation groups, translations, permutation groups) and it was the achievement of Galois to distill the most important axioms (=properties) of groups that turn out to be applicable in many different settings. We will discuss many examples of groups in this class! The class is primarily based on Chapters 1-7 of Artin's book.

1. Group theory
definition of a group, examples (such as the permutation group, GL_n over finite fields, cyclic group, dihedral group), subgroups, homomorphisms, isomorphisms, cosets, products of groups, quotient groups, modular arithmetic

2. Symmetries
orthogonal matrices and rotations, symmetry of plane figures, group of motions of the plane, finite group of motions, discrete groups of motion/wallpaper patterns

3. Group actions
group operations, operation of cosets, counting formula, Burnside formula, finite subgroups of the rotation group, operation of groups on themselves, class equations, operations on subsets, Sylow theorems, groups of order 12, symmetric group, free group, generators and relations

Other material

Group tables
Article on the game "Set"
Group Theory in Sage
Wallpaper groups
Wallpaper groups in real life