Problem 1 - Algebra: Pure and Applied

Question

In Exercises 1 through 10 find the order of the indicated element in the indicated group. $$ 2 \in \mathbb{Z}_{3} $$

Step-by-Step Solution

Answer

The order of 2 in \( \mathbb{Z}_3 \) is 3.

Step 1: Understanding the Group

The group \( \mathbb{Z}_3 \) consists of integers under addition modulo 3. The elements of this group are \( \{0, 1, 2\} \). In this context, we are tasked with finding the order of the element 2 in this group.

Step 2: Definition of Order

The order of an element \( a \) in a group is the smallest positive integer \( n \) such that \( na \equiv 0 \pmod{3} \). In other words, we want to find the smallest \( n \) for which repeated addition of the element equals 0 modulo 3.

Step 3: Calculate Powers of Element

Let's start calculating powers of the element 2 in \( \mathbb{Z}_3 \). We have: - \( 1 imes 2 = 2 \equiv 2 ot\equiv 0 \pmod{3} \)- \( 2 imes 2 = 4 \equiv 1 ot\equiv 0 \pmod{3} \)- \( 3 imes 2 = 6 \equiv 0 \equiv 0 \pmod{3} \)

Step 4: Determine the Order

The smallest positive integer \( n \) for which \( n imes 2 \equiv 0 \pmod{3} \) is 3. Therefore, the order of the element 2 in \( \mathbb{Z}_3 \) is 3.

Key Concepts

Order of Element

In group theory, one significant concept is the "order of an element". It measures how many times you must "add" or "apply" the group operation to an element until you reach the group's identity element. For a group like \( \mathbb{Z}_3 \), which involves addition, the identity element is 0.

To find the order, you start by repeatedly applying the group's operation to the element. For instance, if we take the element '2' in the group \( \mathbb{Z}_3 \), we check a series of sums:

• 2 stands alone, but 2 alone is not 0 mod 3.
• Add 2 to itself (i.e., \(2 + 2 = 4\)) which reduces to 1 mod 3, still not 0.
• Finally, add another 2 (\(2 + 2 + 2 = 6\)), which equals 0 mod 3.

Thus, the order of an element is the smallest number at which the operation results in the identity element. In our case, this number is 3. So, the order of 2 in \( \mathbb{Z}_3 \) is 3 because adding 2, three times brings us back to 0 mod 3.

Modulo Arithmetic

Modulo arithmetic is sometimes called modulus or clock arithmetic. It's like counting on a clock where once you reach the highest number, you start over from zero. This "wrapping around" behavior is very useful.

When you perform arithmetic in a modulo system, your results are reduced under a certain number, called the modulus. For instance, with addition modulo 3, like in \( \mathbb{Z}_3 \), any result that equals or exceeds 3 "wraps" back around. So:

• 3 mod 3 is 0 because 3 divided by 3 leaves a remainder of 0.
• 4 mod 3 is 1 because 4 = 3 * 1 + 1, leaving a remainder of 1.
• Likewise, 5 mod 3 equals 2.

Understanding modulo arithmetic is key for solving problems in many areas of mathematics, including number theory and cryptography, as well as group theory which we're discussing here.

Integers Modulo n

Integers modulo \( n \) is a set of equivalence classes of integers under the equivalence relation of congruence modulo \( n \).

This can be seen as a way to partition the set of all integers into different classes or sets, where members of each set share a common property when reduced by modulo \( n \). In simpler terms:

• For \( \mathbb{Z}_3 \), we work with numbers 0, 1, and 2 because any integer is essentially equivalent to one of these numbers mod 3.
• Each number in \( \mathbb{Z}_3 \) can be considered a complete representative class.
• For larger \( n \), you simply extend this set to include integers: \( 0, 1, 2, \cdots, n-1 \).

The concept of integers modulo \( n \) provides a robust framework to perform arithmetic operations while considering congruence, and it is a foundational tool in various mathematical disciplines, including abstract algebra.