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Group Theory

Explore groups through symmetries, applications, and problems.

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.

Interactive
quizzes

22

Concepts and
exercises

245+
  1. 1

    Introduction

    An introduction to Group Theory through the beauty of symmetry.

    1. Symmetry

      Come to know mathematical groups through symmetry.

    2. Combining Symmetries

      Gain a visual understanding of how groups work.

    3. Group Axioms

      What makes a set into a group?

    4. Cube Symmetries

      Explore group axioms with cube symmetries.

  2. 2

    Fundamentals

    The axioms, subgroups, abelian groups, homomorphisms, and quotient groups.

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      Axioms and Basic Examples

      Dive deeper into groups by exploring some real-world applications.

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      More Group Examples

      See how groups tie into geometry and music.

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      Subgroups

      Learn about the structure of groups within a group.

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      Abelian Groups

      For these groups, composition order doesn't matter.

  3. 3

    Applications

    Number theory, the 15-puzzle, peg solitaire, the Rubik's cube, and more!

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      Number Theory

      Use groups to unlock the secrets of integers.

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      Puzzle Games

      Formulate winning game strategies with groups!

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      Rubik's Cubes

      Apply groups to understand this perplexing toy.

  4. 4

    Advanced Topics

    From the isomorphism theorems to conjugacy classes and symmetric groups.

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      Normal Subgroups

      Explore normality, a critical ingredient in making quotient groups.

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      Isomorphism Theorems

      When are two groups different versions of the same thing?

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      Conjugacy Classes

      Learn a wealth of information about a group by separating its elements by class.

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      The Symmetric Group

      Master the fundamentals of permutations.

  5. 5

    Group Actions

    Burnside's Lemma, semidirect products, and Sylow's Theorems.

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      Group Actions

      Explore the interplay between a group and the set it acts upon.

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      Burnside's Lemma

      Solve challenging counting and combinatorial problems with group theory.

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      Semidirect Products

      Learn a useful technique for building larger groups from smaller ones.

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      Sylow Theorems

      Explore the structure of finite groups and uncover fascinating new relations.

Prerequisites
Next steps