Problem 4 - A Book of Abstract Algebra

Question

What is the order of 1 in \(\mathbb{R}^{*} ?\) What is the order of 1 in \(\mathbb{R} ?\)

Step-by-Step Solution

Answer

The order of 1 in \( \mathbb{R}^{*} \) is 1 and in \( \mathbb{R} \) it is infinite.

Step 1: Understanding the Order in Mathematical Groups

The order of an element in a group is the smallest positive integer \( n \) such that the element raised to the \( n \)-th power equals the identity element of the group. If no such positive integer exists, the order is said to be infinite.

Step 2: Identifying the Relevant Groups

We are considering the order of 1 in two groups: \( \mathbb{R}^{*} \) and \( \mathbb{R} \). \( \mathbb{R}^{*} \) is the multiplicative group of non-zero real numbers, and \( \mathbb{R} \) is the group of real numbers under addition.

Step 3: Determine the Order of 1 in \( \mathbb{R}^{*} \)

In \( \mathbb{R}^{*} \) (multiplicatively), the identity element is 1. Since multiplying 1 by itself any number of times still gives 1, the smallest positive \( n \) doesn't exist that can make it something else. Therefore, the order of 1 is 1.

Step 4: Determine the Order of 1 in \( \mathbb{R} \)

In \( \mathbb{R} \) (additionally), the identity element is 0. The order of 1 is the smallest positive integer \( n \) such that \( 1 + 1 + \ldots + 1 \) (\( n \) times) equals 0. There is no such finite integer, thus the order is infinite.

Key Concepts

Order of an Element

To understand the concept of the order of an element in a group, consider a group as a set with a specific operation that combines any two elements to form another element from the same set. The order of an element is the minimum positive integer \( n \) such that repeatedly applying the group's operation on this element \( n \) times results in the identity element. If no such \( n \) exists, this order is considered infinite. A simple analogy: imagine walking in a circle. If you return to the starting point after a certain number of steps, this number represents the order of your steps. In mathematical groups, this means for an element \( a \), \( a^n = e \) or \( n \times a = 0 \), where \( e \) is the identity element, depending on whether the operation is multiplication or addition, respectively.

Multiplicative Group

A multiplicative group is a set of elements together with the multiplication operation that satisfies four key properties:

• Closure: The product of any two elements in the group is also within the group.
• Associativity: The grouping of elements doesn't change the product (i.e., \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)).
• Identity Element: There exists an element (usually 1) such that multiplying any element by it leaves it unchanged (for any \( a \), \( a \cdot 1 = a \)).
• Inverse: Every element in the group has an inverse such that multiplying the element by its inverse gives the identity element (for any \( a \), there exists \( b \) such that \( a \cdot b = 1 \)).

For example, the set of non-zero real numbers \( \mathbb{R}^* \) constitutes a multiplicative group with these properties. The order of 1 in this group implies the number of times you multiply 1 to get back 1, which is immediately since \(1 \cdot 1 = 1\), meaning the order is 1.

Additive Group

An additive group is similar to a multiplicative group, but with addition as the operation. The properties ensuring it is a group are:

• Closure: The sum of any two elements is also part of the group.
• Associativity: Changing the grouping of additions does not affect the sum (i.e., \((a + b) + c = a + (b + c)\)).
• Identity Element: There exists a 0 such that adding 0 to any element leaves it unchanged (for any \( a \), \( a + 0 = a \)).
• Inverse: Every element has an additive inverse such that their sum is the identity element (for any \( a \), there exists \( b \) such that \( a + b = 0 \)).

The group of real numbers \( \mathbb{R} \) with addition fits this definition. The order of 1 in this context refers to how many times you must add 1 to itself to reach the additive identity, 0. Since repeated addition of 1 will never result in 0, the order of 1 is infinite.

Identity Element

The identity element in a group is a critical component that allows us to define the order of elements. It is the element that "does nothing" when combined with other elements of the group.

• For a multiplicative group, the identity element is typically 1, as multiplying any number by 1 does not change the number (in \( \mathbb{R}^* \), \( a \cdot 1 = a \)).
• For an additive group, the identity is 0, since adding 0 to any number does not change the number (in \( \mathbb{R} \), \( a + 0 = a \)).

The identity element is foundational in determining properties like the order of an element, as it represents the "neutral" outcome of the group operation. Without an identity element, it would be impossible to define the order of an element consistently across the group.