# MATH 150A (Modern Algebra) - Fall 2015, 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

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

## 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 2 in class: pdf

Homework 2: due October 9 in class: pdf

Homework 3: due October 16 in class: pdf

Homework 4: due October 23 in class: pdf (Problems 7, 8 won't be graded since we did not yet reach product groups in class!)

Midterm: October 28, 2015 in class 4:10-5:00pm. The midterm covers Sections 1.4, 1.5, Sections 2.1-2.9, 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 under Resources. Solutions to the homeworks are also available there.
You will only need a pencil and eraser for the test.

No homework due on October 30!

Homework 5: due November 6 in class: pdf

Homework 6: due November 13 in class: pdf

Homework 7: due November 20 in class: pdf

Homework 8: due Wedensday November 25 in class: pdf

Homework 9: due December 4 in class: pdf

Final: Monday, December 7 at 1:00-3:00pm in 216 Wellman.
The final will be comprehensive and cover all topics we discussed in class. More information and practice problems are posted on SmartSite under Resources.