Author(s): Lauren K. Williams
Series: lecture notes
Edition: version 2019-08-22
Year: 2019
Language: English
Commentary: Downloaded from https://integral-domain.org/lwilliams/math280/
I Prerequisites
Sets
Subsets
Set Operations
Partitions
Sets of Numbers
Properties of Real Numbers
Relationships of Real Numbers
Important Theorems
Functions
Relations on Sets
Definition of Function
Injective Functions
Surjective Functions
Bijections
Cardinality of Sets
Equivalence Relations
Equivalence Classes
Modular Arithmetic
Equivalence Modulo n
Reducing Modulo n
Multiplicative Inverses Modulo n
Equivalence Classes
The Integers Modulo n
Matrices
Matrix Arithmetic
Determinants
Inverses
Special Families of Matrices
II Groups
Preliminary Examples
Roots of Unity
The Integers Modulo n
Symmetries of Polygons
Definitions
Binary Operation
Groups
Abelian
Examples
Additive vs Multiplicative Groups
Order of a Group
Trivial Group
Vector Spaces
Basic Group Properties
Why Group Properties Matter
Notation
Uniqueness of the Identity
Uniqueness of Inverses
Involution of Inverses
Shoes-and-Socks Property
Cancellation Laws
Cayley Tables
Group Properties
Ordering Rows and Columns
Constructing Abstract Cayley Tables
Order
Order of a Group
Order of an Element
Subgroups
Definitions
Subgroup Tests
Cyclic Subgroups
Intersections
Centers and Centralizers
Center of a Group
Centralizer of an Element
Cyclic Groups
Finite Cyclic Groups
Subgroups of Cyclic Groups
Infinite Cyclic Groups
Group Homomorphisms
A Review of Functions
Definition
Domain, Codomain, Image, and Kernel
Some Important Homomorphisms
Properties
Group Isomorphisms
Review of Bijective Functions
Definition
Automorphism
Inner Automorphisms
Properties
Other Types of Homomorphisms
Properties of Isomorphic Groups
Cyclic Groups
Permutations
Notation
Cycle Type
Composition
Inverses and Identity
Parity
The Symmetric Group
The Alternating Group
Order of Elements
Cayley's Theorem
Cosets
Properties of Cosets
Index of a Subgroup
Lagrange's Theorem
Useful Corollaries
External Direct Products
Order Properties
Cyclic Groups as Direct Products
Fundamental Theorem of Finite Abelian Groups
Normal Subgroups
Predicting Normal Subgroups
Factor Groups
Simple Groups
Classification of Finite Simple Groups
Monster Group Trivia
How Many Groups Are There?
Other Algebraic Structures
III Symmetry
What is Symmetry?
Planar Transformations
Linear Transformations
Affine Transformations
Isometries
Translations
Rotations
Reflections
Glide Reflections
Fixed Points
Orientation
Finite Symmetry Groups
Rotational Symmetries
Reflective Symmetries
Asymmetry
Examples
Cyclic Symmetry Groups
Dihedral Symmetry Groups
Infinite Symmetry Groups
Da Vinci's Classification Theorem
Proof
Tilings of the Plane
Lattices of the Plane
The Eleven Uniform Tilings of the Plane
Cell Structure
Fundamental Domain
Frieze Groups
The 7 Frieze Groups
Wallpaper Groups
Possible Symmetries
Crystallographic Notation
Orbifold Notation
Wallpaper Group Structures
Symmetry in Other Dimensions and Geometries
The Crystallographic Groups
Spherical Tilings
Hyperbolic Tilings
Group Actions
Classifying Group Actions
Orbits and Stabilizers
