## Introduction

A group is an algebraic structure consisting of a set, together with an associative binary operation, which contains an inverse for each element and an identity element. The operation must satisfy group axioms: closure, associativity, identity and invertibility.

## Axioms

A group is a set G together with a binary operation + such that the following axioms hold.

### Additive Axioms

## Use

### Example

The set of all integers under the operation of addition is a group.

### Example

The set of all non-zero integers under the operation of multiplication is a group.

### Example

The set of twelve elements 0, 1, . . ., 11 under the operation of addition modulo 12 is a group.

This represents the familiar idea of 'clock arithmetic'.

### Example

The set of all invertible square matrices, of a given order, under the operation of matrix multiplication is a group.

The identity matrix is the square matrix, such that each element is 0, other than the elements on the main diagonal, which are 1.