Group Theory
Explore groups through symmetries, applications, and problems.
Symmetry
Combining Symmetries
Group Axioms
Cube Symmetries
Axioms and Basic Examples
More Group Examples
Subgroups
Abelian Groups
Homomorphisms
Quotient Groups
Number Theory
Puzzle Games
Rubik's Cubes
Normal Subgroups
Isomorphism Theorems
Conjugacy Classes
The Symmetric Group
Signs of Permutations
Group Actions
Burnside's Lemma
Semidirect Products
Sylow Theorems
Course description
This course was written in collaboration with Jason Horowitz, who received his mathematics PhD at UC Berkeley and was a founding teacher at the mathematics academy Proof School. This course explores group theory at the university level, but is uniquely motivated through symmetries, applications, and challenging problems. For example, before diving into the technical axioms, we'll explore their motivation through geometric symmetries. You'll be left with a deep understanding of how group theory works and why it matters.
Topics covered
- Abelian Groups
- Conjugacy Classes
- Direct and Semidirect Products
- Group Actions
- Group Axioms
- Homomorphisms
- Isomorphisms
- Normal Groups
- Quotient Groups
- Subgroups
- Sylow Theorems
- Symmetry
Prerequisites and next steps
Familiarity with linear algebra basics and a willingness to engage with mathematical abstraction is all that’s required!