Group Theory Basics

A group is a set of elements with an operation that combines any two to form a third, satisfying four properties.

The Axioms

1. Closure: combining two elements gives another in the set
2. Associativity: (a * b) * c = a * (b * c)
3. Identity: there exists an element e where a * e = a
4. Inverse: every element has an inverse

Examples

• Integers under addition form a group
• Rotations of a square form a group
• Permutations of a set form a group

Group theory is the language of symmetry. It appears everywhere from particle physics to puzzle solving.