# MATH 150A (Modern Algebra) - Fall 2014, UC Davis

## 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

Article on the game "Set"
Group Theory in Sage

## Problem sets

Homework 0: (voluntary) send me an email about yourself, your goal and expectations for the class or anything else you would like to share!

Homework 1: due October 10 in class: pdf

Homework 2: due October 17 in class: pdf

Homework 3: due October 24 in class: pdf

Homework 4: due October 31 in class: pdf

Midterm: November 5, 2014 in class 4:10-5:00pm. The midterm covers Sections 1.4, 1.5, Sections 2.1-2.10, and Section 3.2 of Artin and is based on Homeworks 1-4. There might be a question asking you to state a definition or theorem/proposition. There will also be a question with True/False questions. The other questions will be similar to the homework problems, asking you to prove a particular statement or to give or work out an example. Make sure you are familiar with all the main definitions and theorems (groups, subgroups, normal subgroups, homomorphisms, isomorphisms, cosets, index of a subgroup, Lagrange theorem, ....) and examples of groups (symmetric group, dihedral group, matrix groups over a field, ....). A sample midterm is posted on SmartSite.
You will only need a pencil and eraser for the test.

No homework due on November 7!

Homework 5: due November 14 in class: pdf

Homework 6: due November 21 in class: pdf

Homework 7: due Wednesday November 26 in class: pdf

Homework 8: due December 5 in class: pdf

Homework 9: due December 12 in class: pdf

Final: December 17, 2014 3:30-5:30pm in the usual class room! A file with information on the final and some practice problems can be found under Resources on SmartSite.