Group Theory
A 31-module path through group theory, in study order from foundations to the frontier.
Modules
- Binary Operations and the Group Axioms
- Basic Consequences of the Group Axioms
- Examples: Cyclic Groups and Modular Arithmetic
- Symmetric Groups and Permutation Notation
- Dihedral Groups
- Matrix Groups: GL, SL, O, and U
- Subgroups and the Subgroup Criterion
- Order of an Element
- Cyclic Groups and Generators
- Structure and Classification of Cyclic Groups
- Cycle Decomposition and Parity
- Cosets and Lagrange's Theorem
- Centralizers, Normalizers, and the Center
- Normal Subgroups
- Quotient Groups
- Group Homomorphisms
- Group Isomorphisms
- The Isomorphism Theorems
- Group Actions
- The Orbit-Stabilizer Theorem
- The Class Equation
- Cauchy's Theorem
- p-Groups and Their Properties
- The Sylow Theorems
- Direct Products
- Semidirect Products
- Fundamental Theorem of Finite Abelian Groups
- The Alternating Group
- Simple Groups
- Composition Series and the Jordan-Holder Theorem
- Solvable and Nilpotent Groups