Number Theory Part 1: Group Theory

In continuation with my previous blogs, let’s begin by discussing a group.

Group

A group is a mathematical structure, denoted by {G, ◦} where G is the set of elements and ◦ the operator that consists of a set of elements together with a binary operation that satisfies four fundamental properties:

A1. Closure: The operation must be closed, which means that if a and b are in the set, the element a ◦ b = c is also in the set. The symbol ◦ denotes the operator for the desired operation.

A2. Associativity: The operation must be associative, which means that

(a ◦ b) ◦ c = a ◦ (b ◦ c)

A3 Identity element: There must be an identity element in the group. An element i would be called an identity element if, for every a in the set, we have a ◦ i = a.

A4. Inverse element: Every element in the group must have an inverse element. For every a in the set, the set must also contain an element b such that a ◦ b = i.

Let’s understand through examples.

1. The set of integers under addition: The set of integers is closed under addition, which means the addition of any two integers is also an integer. The addition operation is associative, which means the order in which we add integers does not matter. The identity element for addition is 0 because 0 added to any integer is that integer. Every integer has an inverse element because, for every integer a, there is an integer −a such that a+(−a)=0.
2. The set of integers under multiplication: The set of integers is closed under multiplication, which means the multiplication of any two integers is also an integer. The multiplication operation is associative, which means the order in which we multiply integers does not matter. The identity element for addition is 1 because 1 multiplied by any integer is that integer. The set of integers under multiplication does not contain an inverse element or an integer (it may be a real number).
3. The set of real numbers under multiplication: The set of real numbers is closed under multiplication, which means the multiplication of any two real numbers is also a real number. The multiplication operation is associative, which means the order in which we multiply real numbers does not matter. The identity element for multiplication is 1, because 1 multiplied by any real number is that number. Every real number has an inverse element because, for every real number a, there is a real number a⁻¹such that a*a⁻¹=1

Abelian Group

An abelian group is a commutative group where the order of elements doesn’t affect the operation outcome.

An Abelian group’s properties include A1-A4, which group theory discusses. They also have another property.

A5. Commutativity: An operation ◦ is commutative if

a ◦ b = b ◦ a

Let me allow you to demonstrate with examples.

1. The set of integers under addition: The set of integers is closed under addition, is associative, has an identity element for addition, which is 0, and every integer has an inverse element. The group operation is commutative because a+b=b+a for all integers a and b.
2. The set of real numbers under multiplication: The set of real numbers is closed under multiplication, is associative, has an identity element for addition, which is 1, and every integer has an inverse element. The group operation is commutative because a×b=b×a for all real numbers a and b.

In summary, groups and abelian groups are essential mathematical structures with various applications in various fields, such as cryptography, physics, and coding theory. They offer a framework for comprehending and resolving complex issues by capturing vital symmetry, structure, and interaction properties. In summary, these structures are fundamental mathematics tools that significantly impact many areas of study.

Number Theory Part 2: Ring

Number Theory Part 3: Integral Domain

Number Theory Part 4: Fields