Algebra I: Lecture Notes

Groups
Introduction
Laws of Composition
Permutation and Symmetric Groups
Examples of Symmetric Groups
Subgroups and Cyclic Groups
Review
Subgroups
Subgroups of the Integers
Cyclic Groups
Homomorphisms and Isomorphisms
Review
Homomorphisms
Examples
Isomorphisms and Cosets
Review
Isomorphisms
Automorphisms
Cosets
Lagrange's Theorem
The Correspondence Theorem
Review
Lagrange's Theorem
Results of the Counting Formula
Normal Subgroups
The Correspondence Theorem
Normal Subgroups and Quotient Groups
Review
Normal Subgroups
Quotient Groups
First Isomorphism Theorem
Fields and Vector Spaces
Review
Fields
Vector Spaces
Bases and Dimension
Dimension Formula
Review
Matrix of Linear Transformations
Dimension Formula
Dimension Formula
Review
Linear Operators
Change of Basis
Eigenvectors, Eigenvalues, and Diagonalizable Matrices
Finding Eigenvalues and Eigenvectors
Eigenbases and the Jordan Form
Review
The Characteristic Polynomial
Jordan Form
The Jordan Decomposition
Review
The Jordan Decomposition, Continued
Proof of Jordan Decomposition Theorem
Orthogonal Matrices
Dot Products and Orthogonal Matrices
The Special Orthogonal Group
Orthogonal Matrices in Two Dimensions
Orthogonal Matrices in Three Dimensions
Isometries
Review
Isometries
Isometries in 2-space
Symmetry Groups
Review
Examples of Symmetry Groups
Discrete Subgroups of R
Finite subgroups of O2
More Discrete Subgroups
Finite and Discrete Subgroups, Continued
Review
Finite Subgroups of M2
Discrete Subgroups of M2
Discrete Subgroups of R2
Back to Discrete Subgroups of M2!
Discrete Groups
Review
Examples for L and G
Crystallographic Restriction
Group Actions
Review
Motivating Examples
What is a group action?
The Counting Formula
Stabilizer
Review
Counting Formula
Stabilizers of Products
Statement
Finding the subgroups
The Octahedral Group
Group Actions on G
Conjugation
p-groups
The Icosahedral Group
Review: The Class Equation
Basic Information
Conjugacy Classes
Simple Groups
Conjugacy Classes for Symmetric Groups
Conjugacy Classes for Symmetric and Alternating Groups
Review
Cycle Type
Conjugacy Classes in Sn
Class Equation for S4
Student Question
The Sylow Theorems
Review
Motivation
The First Sylow Theorem
The Second Sylow Theorem
The Third Sylow Theorem
Applications of the Sylow Theorems
Proofs and Applications of the Sylow Theorems
Review
Application: Decomposition of Finite Abelian Groups
Proof of Sylow Theorems
Bilinear Forms
Review
Bilinear Forms
Change of Basis
Bilinear Forms over C
Orthogonality
Review: Bilinear Forms
Hermitian Forms
Orthogonality
The Projection Formula
Review: Symmetric and Hermitian Forms
Orthogonality
Orthogonal Bases
Projection Formula
Euclidean and Hermitian Spaces
Review: Orthogonal Projection
Euclidean and Hermitian Spaces
Gram-Schmidt Algorithm
Complex Linear Operators
The Spectral Theorem
Review: Hermitian Spaces
The Spectral Theorem
Linear Groups
Geometry of groups
Geometry of SU2
Quaternions
Geometry of the Sphere
Latitudes
The Special Unitary Group SU2
Review
Longitudes
More Group Theoretic Properties
Conjugation and the Orthogonal Group
One-Parameter Groups
One-Parameter Subgroups
Review
Properties of the Matrix Exponential
One-Parameter Subgroups
One-Parameter Groups, Continued
Review
Examples!
The Special Linear Group SLn(C)
Tangent Vectors
Lie Groups
Review
Lie Groups
Manifolds
Lie Bracket
Simple Linear Groups
Review
Simple Linear Groups
The Special Unitary Group
The Special Linear Group
Generalizations
Hilbert's Third Problem
Polygons in the Plane
The Question
Some Algebra
Back to Polytopes