# MATH 150A (Modern Algebra) - Fall 2011, 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 1-6 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

Permutations

## Other material

Article on the game "Set"

## 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 September 30, 2011 in class: pdf
Solution: pdf (written by Mark and Nathan as an example on how to write up your solutions!)

Homework 2: due October 7, 2011 in class: pdf
Solution: pdf (written by Mark and Nathan as an example on how to write up your solutions!)

Homework 3: due October 14, 2011 in class: pdf
Solution: pdf (written by Mark and Nathan as an example on how to write up your solutions!)

Homework 4: due October 21, 2011 in class: pdf
Solution: pdf (written by Mark and Nathan as an example on how to write up your solutions!)

Homework 5: due October 28, 2011 in class: pdf
Solution: pdf (written by Mark and Nathan as an example on how to write up your solutions!)

Midterm: October 28, 2011 in 3 Kleiber 10:00-10:50am.
The midterm covers Sections 1.3, 1.4, all of Section 2, and Section 3.2 of Artin and is based on Homeworks 1-5.
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 presentations of permutations (like two-line notation, one-line notation, cycle notation, permutation matrices) and that you can easily convert between them. Make sure you remember how to compose permutations, compute the sign of a permutation, how to determine whether two permutations are conjugate etc.. You will only need a pencil and eraser for the test.
Practice midterm ( solutions )
Midterm Solution

No Homework due November 4, 2011.

Homework 6: due November 9, 2011 in class (Friday Nov 11 is a holiday): pdf
Solution: pdf

Homework 7: due November 18, 2011 in class: pdf
Solution: pdf

Take Home Midterm: will be handed out November 21, 2011 in class; due November 23, 2011 in class: pdf
Solution: pdf

Homework 8: due December 2, 2011 in class: pdf (last one!)
Solution: pdf

Final: December 5, 2011 from 10:30am-12:30pm in 1001 Giedt
The final is comprehensive and covers all material we discussed during class. Here is some further information and practice problems that will be discussed during the discussion session on Thursday.
Practice problems