# MATH 150A (Modern Algebra) - Winter 2023, 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