Group Theory

Introduction

A group is a *set* of *elements* equipped with a binary operator that satisfies the following properties:

• Closure
• Identity Element
• Every element has an inverse
• Associativity

Binary Operator

A binary operator is an operator that takes two operands to perform an operation (or calculation), e.g. + - / * %.

Example: a . b a and b are the *operands* "." is the *operator*

Examples of binary operators:

• Arithmetic Operators: + -
• Comparison Operators: < > =  !=
• Boolean Operators: AND, OR, XOR

Closure

The operation must be closed. When the operator is applied to any two elements in the set, the result must also be an element in the set.

Closure works for both finite and infinite sets.

Example using a finite set:

 ℤ₅ = {0,1,2,3,4}

Operation = addition mod 5 Test closure:

 3 + 4 = 7 → 7 mod 5 = 2 (in the set)  
 4 + 4 = 8 → 8 mod 5 = 3 (in the set)  
 1 + 3 = 4 (in the set)

Identity Element

• The identity element I must be unique.
• There cannot be more than one.
• There must exist an element I in the set such that for every element a:

     a . I = I . a = a

Examples:

• Integers under addition → identity is 0

     a + 0 = 0 + a = a

• Integers under multiplication → identity is 1

     a * 1 = 1 * a = a

Inverse

• Every element in the set must have an inverse.
• For each element a in the set, there must exist an element a₍ᵢₙᵥ₎ such that:

     a . a₍ᵢₙᵥ₎ = a₍ᵢₙᵥ₎ . a = I

Example (integers under addition):

     a + (-a) = (-a) + a = 0
     5 + (-5) = (-5) + 5 = 0

Associativity

Associativity ensures that the result of applying the operation does *not* depend on how the elements are parenthesised.

     (a . b) . c = a . (b . c)