Group Axioms

Introduction

A group is an ordered pair $(G, \cdot)$ , where $G$ is a non-empty set and $\cdot$ is a binary operation on $G$ , satisfying the following four axioms.

The Four Group Axioms

1. Closure

For all $a, b \in G$ , the result of the operation, $a \cdot b$ , is also in $G$ .

2. Associativity

For all $a, b, c \in G$ , the equation $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ holds.

3. Identity Element

There exists a unique element $e \in G$ , called the identity element, such that for every element $a \in G$ , the equation $e \cdot a = a \cdot e = a$ holds.

4. Inverse Element

For each element $a \in G$ , there exists an element $b \in G$ , denoted $a^{- 1}$ , called the inverse of $a$ , such that $a \cdot a^{- 1} = a^{- 1} \cdot a = e$ .

Additional Properties

Commutativity

If the group operation is also commutative (i.e., $a \cdot b = b \cdot a$ for all $a, b \in G$ ), the group is called an abelian group or a commutative group.

Order

The number of elements in a group is called its order, denoted $| G |$
If the order is finite, the group is a finite group
If the order is infinite, the group is an infinite group

Motivation: Symmetry

The group axioms are not an arbitrary collection of rules; rather, they are a precise formalization of the essential properties common to all systems of symmetry transformations.

Consider the set of symmetries of a geometric object:

The composition of two symmetries is another symmetry (Closure)
The composition of functions is inherently associative (Associativity)
The transformation that does nothing is a symmetry (Identity)
Any symmetry can be undone by reversing the transformation (Inverse)

Thus, the set of symmetries of any object forms a group, known as its symmetry group. This insight frames group theory as the abstract study of symmetry itself.

Examples

Example 1: The Integers under Addition

$(Z, +)$ is a group:

Closure: The sum of two integers is an integer
Associativity: $(a + b) + c = a + (b + c)$ for all $a, b, c \in Z$
Identity: $0$ is the identity element
Inverses: The inverse of $a$ is $- a$
Abelian: Addition is commutative

Example 2: The Non-zero Rationals under Multiplication

$(Q^{*}, \cdot)$ is a group:

Closure: The product of two non-zero rationals is non-zero rational
Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in Q^{*}$
Identity: $1$ is the identity element
Inverses: The inverse of $a$ is $\frac{1}{a}$
Abelian: Multiplication is commutative

Example 3: The Symmetric Group $S_{3}$

The group of permutations of three elements:

Closure: Composition of permutations is a permutation
Associativity: Function composition is associative
Identity: The identity permutation
Inverses: Every permutation has an inverse
Non-abelian: Permutation composition is not commutative

Basic Properties

Uniqueness of Identity

The identity element in a group is unique.

Uniqueness of Inverses

For each element $a \in G$ , the inverse $a^{- 1}$ is unique.

Cancellation Laws

In a group, if $a b = a c$ , then $b = c$ (left cancellation), and if $b a = c a$ , then $b = c$ (right cancellation).

Inverse of a Product

For any elements $a, b \in G$ , we have $(a b)^{- 1} = b^{- 1} a^{- 1}$ .