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Brilliant

Group Theory

via Brilliant

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Overview

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.

Syllabus

  • Introduction: An introduction to Group Theory through the beauty of symmetry.
    • Symmetry: Come to know mathematical groups through symmetry.
    • Combining Symmetries: Gain a visual understanding of how groups work.
    • Group Axioms: What makes a set into a group?
    • Cube Symmetries: Explore group axioms with cube symmetries.
  • Fundamentals: The axioms, subgroups, abelian groups, homomorphisms, and quotient groups.
    • Axioms and Basic Examples: Dive deeper into groups by exploring some real-world applications.
    • More Group Examples: See how groups tie into geometry and music.
    • Subgroups: Learn about the structure of groups within a group.
    • Abelian Groups: For these groups, composition order doesn't matter.
    • Homomorphisms: Play with group functions that preserve the groups' structures.
    • Quotient Groups: It's a bit like factoring groups...
  • Applications: Number theory, the 15-puzzle, peg solitaire, the Rubik's cube, and more!
    • Number Theory: Use groups to unlock the secrets of integers.
    • Puzzle Games: Formulate winning game strategies with groups!
    • Rubik's Cubes: Apply groups to understand this perplexing toy.
  • Advanced Topics: From the isomorphism theorems to conjugacy classes and symmetric groups.
    • Normal Subgroups: Explore normality, a critical ingredient in making quotient groups.
    • Isomorphism Theorems: When are two groups different versions of the same thing?
    • Conjugacy Classes: Learn a wealth of information about a group by separating its elements by class.
    • The Symmetric Group: Master the fundamentals of permutations.
    • Signs of Permutations: Connect the Legendre symbol from number theory to permutation signs.
  • Group Actions: Burnside's Lemma, semidirect products, and Sylow's Theorems.
    • Group Actions: Explore the interplay between a group and the set it acts upon.
    • Burnside's Lemma: Solve challenging counting and combinatorial problems with group theory.
    • Semidirect Products: Learn a useful technique for building larger groups from smaller ones.
    • Sylow Theorems: Explore the structure of finite groups and uncover fascinating new relations.

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