Problem 2 - A Book of Abstract Algebra

Question

Every complex number \(a+b \mathbf{i}\) may be represented as a point in the complex plane. The unit circle in the complex plane consists of all the complex numbers whose distance from the origin is 1 ; thus, clearly, the unit circle consists of all the complex numbers which can be written in the form $$ \cos x+\mathbf{i} \sin x $$ for some real number \(x\). Let \(T\) designate the set \(\\{\) cis \(x: x \in \mathbb{R}\\}\), that is, the set of all the complex numbers lying on the unit circle, with the operation of multiplication. Use part 1 to prove that \(T\) is a group. ( \(T\) is called the circle group.)

Step-by-Step Solution

Answer

The set \( T \) under multiplication is a group as it satisfies closure, associativity, identity, and inverses.

Step 1: Define Group Criteria

To prove that a set with an operation is a group, we must show it satisfies four properties: closure, associativity, identity element, and inverse elements.

Step 2: Show Closure Property

The closure property requires that the product of any two elements in the set is also in the set. Let two arbitrary elements on the unit circle be \(z_1 = \cos x_1 + \mathbf{i} \sin x_1\) and \(z_2 = \cos x_2 + \mathbf{i} \sin x_2\). Then, \( z_1z_2 = (\cos x_1 + \mathbf{i} \sin x_1)(\cos x_2 + \mathbf{i} \sin x_2) \).Expanding this using the formula for the product of complex numbers and the angle addition formulas, we find:\[ z_1z_2 = \cos(x_1 + x_2) + \mathbf{i} \sin(x_1 + x_2) \]Since \( \cos(x_1 + x_2) + \mathbf{i} \sin(x_1 + x_2) \) is another element of \(T\), the set is closed under multiplication.

Step 3: Show Associativity Property

Multiplication of complex numbers is associative, similar to real numbers. This means for any three complex numbers \(z_1, z_2, z_3\) in \(T\), we have:\[ (z_1 z_2) z_3 = z_1 (z_2 z_3) \]Thus, the operation of multiplication on \(T\) is associative.

Step 4: Identify the Identity Element

The identity element in a group is an element such that multiplying any element of the group by it leaves the element unchanged.In the circle group \(T\), the identity element is \(z = 1 = \cos 0 + \mathbf{i} \sin 0\).For any \(z = \cos x + \mathbf{i} \sin x\) in \(T\),\[ z \cdot 1 = (\cos x + \mathbf{i} \sin x)(1) = \cos x + \mathbf{i} \sin x \]Thus, the identity element is \(1\) and exists in \(T\).

Step 5: Find Inverse Elements

For every element \(z = \cos x + \mathbf{i} \sin x\) in \(T\), we need to find an element \(z^{-1}\) such that \(z \cdot z^{-1} = 1\).The inverse of \(z\) is \(z^{-1} = \cos(-x) + \mathbf{i} \sin(-x)\). This is because:\[ z \cdot z^{-1} = (\cos x + \mathbf{i} \sin x)(\cos(-x) + \mathbf{i} \sin(-x)) = \cos(0) + \mathbf{i} \sin(0) = 1 \]Since the inverse element \(z^{-1}\) also lies on the unit circle, it is in \(T\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real number system that includes all numbers of the form \(a + b \mathbf{i}\). Here, \(a\) and \(b\) are real numbers, and \(\mathbf{i}\) is the imaginary unit, which satisfies \(\mathbf{i}^2 = -1\). Complex numbers can be represented as points in the complex plane, where the horizontal axis is the real axis and the vertical axis is the imaginary axis.
The notation \(a + b \mathbf{i}\) describes a point in this two-dimensional plane. For example, the number \(3 + 2 \mathbf{i}\) would be located 3 units along the real axis and 2 units along the imaginary axis. This visualization helps us understand operations like addition and multiplication, where each operation has a geometrical interpretation.
Multiplied complex numbers result in rotation and scaling of the original vector. This makes complex numbers a powerful tool in various fields such as engineering, physics, and applied mathematics.

Group Theory

In mathematics, group theory is a branch that studies the algebraic structures known as groups. A group is a set combined with an operation that satisfies four fundamental properties:

• Closure: If you take any two elements from the set and apply the operation, the result should be another element of the set.
• Associativity: The order in which you perform the operation doesn't change the result. For elements \(a, b, c\) in the group, \((ab)c = a(bc)\).
• Identity element: There is an element in the group such that when you combine it with any other element, the latter remains unchanged.
• Inverse elements: For every element in the group, there is an inverse such that their combination gives the identity element.

In the context of the unit circle, the set of complex numbers forming the circle group, with multiplication as the operation, satisfies these group criteria, making it a compelling example of group theory in action.

Unit Circle

The unit circle is a crucial concept in mathematics, particularly in trigonometry and complex analysis. It is defined in the complex plane as the set of all points (complex numbers) that are exactly one unit away from the origin. Points on the unit circle can be expressed in the form \(\cos x + \mathbf{i} \sin x\), where \(x\) is a real number corresponding to the angle in radians from the positive real axis.
This representation is significant because any complex number on the unit circle has an absolute value of 1. When working with complex numbers, this simplifies calculations involving multiplication and division, as magnitude does not need further computation.
The unit circle also ties neatly into trigonometric functions, as it provides a geometric interpretation of sine and cosine for any angle \(x\). This creates a powerful link between algebraic and geometric perspectives, which is foundational for many applications across scientific domains.

Closure Property

The closure property implies that performing an operation (like multiplication) on any two elements of a set should yield another element within the same set. For a group, closure must be proven to establish the group structure.
Within the circle group, given two elements \(z_1 = \cos x_1 + \mathbf{i} \sin x_1\) and \(z_2 = \cos x_2 + \mathbf{i} \sin x_2\), their product \(z_1z_2\) must also lie on the unit circle. Using trigonometric identities to multiply these expressions, we obtain:
\[ z_1z_2 = \cos(x_1 + x_2) + \mathbf{i} \sin(x_1 + x_2) \]
This result confirms that the product remains within the unit circle, demonstrating the closure property. This property is critical in showing that the set forms a group under multiplication, as every resulting product of two initial elements stays within the boundaries set by the unit circle.