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 a type of operator that takes tow operands to perform an operation (or calculation) e.g. + - / * %

e.g a . b

a and b are the operands
. is the operator
So we have

• Arithmetic Operators + -
• Comparison Operators < > = !=
• Boolean Operators AND, OR, XOR, NOT

Closure

The operation must be closed. When the operator is applied to any two elements in the set the result must also be and element in the set. So this can work on infinite and finite sets given

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 identity element I in the set such that for every element in a in the set

a . I = I . a = a

For example Integers under multiplication I = 1 where a = 5

a * 1 = 1 * a = 1 = a
5 * 1 = 1 * 5 = 1 = 5

The identity element I must be unique There cannot be more than one

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

For example: Integers under addition where a =5

   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 parenthesitised.
For example

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