Group Theory

Explore groups through symmetries, applications, and problems.


Combining Symmetries

Group Axioms

Cube Symmetries

Axioms and Basic Examples

More Group Examples


Abelian Groups


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!


  • Number Theory
  • Complex Numbers