---
title: Groups
description: Groups are algebraic structures consisting of a set and a single operation that satisfies closure, associativity, identity, and invertibility. Learners will understand how to analyze symmetry, work with subgroups, and apply group theory to abstract mathematical problems.
category: mathematics
subcategory: abstract-algebra
difficulty: beginner, intermediate, advanced
url: /subject/groups
---

# Groups

Groups are algebraic structures consisting of a set and a single operation that satisfies closure, associativity, identity, and invertibility. Learners will understand how to analyze symmetry, work with subgroups, and apply group theory to abstract mathematical problems.

## Available Resources

1 Videos • 2 Books • 2 Courses • 1 Websites

## Videos

### 1. Euler's Formula with Introductory Group Theory

**Author:** Grant Sanderson

3Blue1Brown video introducing groups through symmetry actions, then using the multiplicative and additive groups of numbers to explain why exponentials turn addition into multiplication and what Euler's formula means geometrically, building visual intuition for group actions and homomorphisms.

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

**Link:** https://www.youtube.com/watch?v=mvmuCPvRoWQ

**Tags:** group-theory, symmetry-groups, homomorphisms, eulers-formula, visual-math

## Courses

### 1. Introduction to Groups, Rings and Fields

Learn group theory basics with N J Wildberger's "Introduction to Groups, Rings and Fields" course. Start your abstract algebra journey today!

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

**Link:** https://www.youtube.com/playlist?list=PLyqSpQzTE6M8aLVVwxkJ44abrFFzdioId

**Tags:** group-theory, abstract-algebra, rings-and-fields, lecture-series, mathematics

### 2. Abstract Algebra

Learn group theory with Prof. Benedict Gross's Abstract Algebra course. Explore fundamental concepts with this accessible video series!

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

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

**Tags:** group-theory, abstract-algebra, rings-and-fields, galois-theory, lecture-series

## Books

### 1. Abstract Algebra

**Author:** David S. Dummit, Richard M. Foote

The standard comprehensive reference for a first and second course in algebra, covering groups, rings, modules, and fields through Galois theory, with enough worked detail and exercises to serve as a self-study source.

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

**Link:** https://www.amazon.com/dp/0471433349?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, abstract-algebra

### 2. A First Course in Abstract Algebra

**Author:** John B. Fraleigh

A gentler first pass through groups, rings, and fields, building each idea from concrete examples before the abstraction, with graded exercises that make it workable without an instructor. Commonly assigned before Dummit and Foote.

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

**Link:** https://www.amazon.com/dp/0201534673?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, abstract-algebra

## Websites

### 1. 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

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