Groups

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

∀ x, y ∈ G, x + y ∈ G ∀ x, y, z ∈ G, x + (y + z) = (x + y) + z ∃ 0 ∈ G, ∀ x ∈ G, x + 0 = x ∀ x ∈ G, ∃ -x ∈ G, x + -x = 0

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.