Math 280 Modern Algebra, Fall 2019

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