Show that the set G = {ω, ω^2, ω^3} is a group with respect to multiplication, where ω is an imaginary cube root of unity.

Introduction

In the context of discrete structures, a group is a set of elements combined with an operation that combines any two of its elements to form a third element in such a way that four conditions called group axioms are satisfied. These conditions are closure, associativity, identity and invertibility.

The imaginary cube root of unity, ω, is a complex number such that ω^3 = 1. The three cube roots of unity are 1, ω, and ω^2. In this case, we are considering the set G = {ω, ω^2, ω^3}.

Group Properties

The four properties that a set must satisfy to be a group are:

1. Closure: For all a, b in G, the result of the operation, a * b, is also in G.
2. Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
3. Identity element: There is an element e in G such that, for every element a in G, the equations e * a and a * e return a.
4. Inverse element: For each a in G, there exists an element b in G, commonly denoted a^−1 (or −a, if the operation is denoted "+"), such that a * b = e and b * a = e.

Closure

The set G is closed under multiplication. This is because the multiplication of any two elements in G results in another element that is also in G. This can be demonstrated using the property ω^3 = 1.

For example, ω * ω^2 = ω^3 = 1 which is in G. Similarly, ω^2 * ω^2 = ω^4 = ω * ω^3 = ω * 1 = ω which is also in G.

Associativity

Multiplication is associative, meaning that for any a, b, c in G, (a * b) * c = a * (b * c).

For example, let a = ω, b = ω^2, and c = ω^3. Then, (ω * ω^2) * ω^3 = ω^3 * ω^3 = 1 * 1 = 1. On the other hand, ω * (ω^2 * ω^3) = ω * 1 = ω. Therefore, multiplication in G is associative.

Identity Element

The identity element in the set G is ω^3 = 1. This is because for any a in G, a * 1 = a and 1 * a = a.

For example, ω * 1 = ω and 1 * ω = ω. Similarly, ω^2 * 1 = ω^2 and 1 * ω^2 = ω^2.

Inverse Element

Each element in G has an inverse in G. This means that for any a in G, there exists b in G such that a * b = 1 and b * a = 1.

For example, the inverse of ω is ω^2 because ω * ω^2 = ω^3 = 1 and ω^2 * ω = ω^3 = 1. Similarly, the inverse of ω^2 is ω because ω^2 * ω = ω^3 = 1 and ω * ω^2 = ω^3 = 1.

Conclusion

In conclusion, the set G = {ω, ω^2, ω^3} satisfies all the properties of a group with respect to multiplication. It is closed under multiplication, multiplication is associative in G, there is an identity element in G, and each element in G has an inverse in G. Therefore, G is a group with respect to multiplication.

Diagram

A diagram is not necessary for this question as all the concepts and proofs can be clearly explained using mathematical notation and examples.

Summary

The set G = {ω, ω^2, ω^3} is a group with respect to multiplication, where ω is an imaginary cube root of unity.

Analogy

Imagine a group of three friends, ω, ω^2, and ω^3, who have a special way of combining their powers. When they multiply their powers together, they form a new element in the group.