# Group Theory

## Explore groups through symmetries, applications, and problems.

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.

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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.

1. Included with

#### Axioms and Basic Examples

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

2. Included with

#### More Group Examples

See how groups tie into geometry and music.

3. Included with

#### Subgroups

Learn about the structure of groups within a group.

4. Included with

#### 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!

1. Included with

#### Number Theory

Use groups to unlock the secrets of integers.

2. Included with

#### Puzzle Games

Formulate winning game strategies with groups!

3. Included with

#### Rubik's Cubes

Apply groups to understand this perplexing toy.

4. 4

From the isomorphism theorems to conjugacy classes and symmetric groups.

1. Included with

#### Normal Subgroups

Explore normality, a critical ingredient in making quotient groups.

2. Included with

#### Isomorphism Theorems

When are two groups different versions of the same thing?

3. Included with

#### Conjugacy Classes

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

4. Included with

#### The Symmetric Group

Master the fundamentals of permutations.

5. 5

### Group Actions

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

1. Included with

#### Group Actions

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

2. Included with

#### Burnside's Lemma

Solve challenging counting and combinatorial problems with group theory.

3. Included with