---
title: Group Actions
description: Group actions describe how algebraic groups operate on sets to transform their elements. Learners will understand how to analyze symmetries, apply Burnside's Lemma, and use orbit-stabilizer theorems to solve problems in geometry, combinatorics, and physics.
category: mathematics
subcategory: abstract-algebra
difficulty: beginner, intermediate, advanced
url: /subject/group-actions
---

# Group Actions

Group actions describe how algebraic groups operate on sets to transform their elements. Learners will understand how to analyze symmetries, apply Burnside's Lemma, and use orbit-stabilizer theorems to solve problems in geometry, combinatorics, and physics.

## Where to start

Start with the Group Properties Wiki, a free wiki covering group actions with examples. If you only use one resource, make it Benedict Gross's Abstract Algebra, an advanced lecture course that treats group actions as part of abstract algebra.

## Available Resources

1 Courses • 2 Websites

## Websites

### 1. Group Properties Wiki

The Groupprops wiki entry on group actions, giving equivalent definitions, related notions such as orbits, stabilizers, transitive and faithful actions, and links to many worked examples and facts. Readers can use it as a precise reference while studying group theory.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://groupprops.subwiki.org/wiki/Group_action

**Tags:** group-actions, group-theory, abstract-algebra, orbits-and-stabilizers, permutation-groups

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Abstract Algebra

Learn group actions with Prof. Benedict Gross! This abstract algebra course explores key concepts with clear explanations.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLzUeAPxtWcqzr80lS25FrzMn7a36BuXhj

**Tags:** abstract-algebra, group-theory, group-actions, ring-theory, linear-algebra

---

*This content is part of Dantes.io - Your Treasure Map to Knowledge*

*Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).*

View this page online: https://dantes.io/subject/group-actions