Problem 2 - Algebra: Pure and Applied

Question

Find the order of the indicated element in the indicated group. $$ 4 \in \mathbb{Z}_{10} $$

Step-by-Step Solution

Answer

The order of 4 in \( \mathbb{Z}_{10} \) is 5.

Step 1: Understand the Context

The problem asks us to find the order of 4 in the group \( \mathbb{Z}_{10} \). \( \mathbb{Z}_{10} \) is the group of integers under addition modulo 10. The order of an element \( a \) in this context is the smallest positive integer \( n \) such that \( n \times a \equiv 0 \pmod{10} \).

Step 2: Identify the Operation

In \( \mathbb{Z}_{10} \), the operation is addition modulo 10. To find the order of an element, we need to repeatedly add the element to itself until we get an equivalent of 0 under modulo 10.

Step 3: Calculate Repeated Additions

Start by checking how many times we need to add 4 to itself to get a multiple of 10:- \( 1 \times 4 = 4 \equiv 4 \pmod{10} \)- \( 2 \times 4 = 8 \equiv 8 \pmod{10} \)- \( 3 \times 4 = 12 \equiv 2 \pmod{10} \)- \( 4 \times 4 = 16 \equiv 6 \pmod{10} \)- \( 5 \times 4 = 20 \equiv 0 \pmod{10} \)

Step 4: Determine the Order

From Step 3, we find that \( 5 \times 4 \equiv 0 \pmod{10} \), and no smaller positive multiple of 4 equals 0 modulo 10. Therefore, the order of 4 in \( \mathbb{Z}_{10} \) is 5.

Key Concepts

Order of an Element

In group theory, the **order of an element** is an important concept. Put simply, the order of an element refers to the smallest number of times you need to combine an element with itself to get back to the group's identity element.

For example, consider the group of integers under addition modulo 10, denoted as \( \mathbb{Z}_{10} \). This group's identity element is 0 because adding 0 to any number doesn't change that number. Here, the **order of the element 4** means finding the smallest positive integer, say "\( n \)", where adding 4 to itself \( n \) times results in 0.

• Start with adding \( n = 1 \): \( 4 + 4 = 8 \) (not 0 yet, so keep going)
• Continue to \( n = 5 \): \( 4 + 4 + 4 + 4 + 4 = 20 \equiv 0 \pmod{10} \)

Thus, the order of 4 in this group is 5 because you need to add 4 to itself five times to get a total sum that is congruent to 0 under modulo 10.

Modulo Arithmetic

**Modulo arithmetic** is like clockwork, which means numbers wrap around after reaching a certain value, known as the modulus. The basic idea is to find the remainder when a number is divided by the modulus.

In our case, let's look at modulo 10. Here, any calculations are handled under this 'clock' of 10 numbers. For example, according to below examples, calculating the modulo essentially gives the remainder:

• Example 1: \( 15 \div 10 \) has a remainder of 5, so \( 15 \equiv 5 \pmod{10} \)
• Example 2: \( 28 \equiv 8 \pmod{10} \)
• Example 3: \( 46 \equiv 6 \pmod{10} \)

This kind of arithmetic is not only crucial in finding the order of elements in groups like \( \mathbb{Z}_{10} \) but is also widely used in computer science for hashing functions and cryptography.

Cyclic Groups

A **cyclic group** is a type of group where you can get every element in the group by repeatedly applying the group operation to a particular element, known as a generator. Cyclic groups are particularly important because they demonstrate a simple structure that can represent more complex group concepts.

For example, in the group \( \mathbb{Z}_{10} \), if you pick the number 1 as the generator, you can create every other element by adding 1 repeatedly:

• 0: the identity element
• 1: = generator itself
• 2: = 1 + 1
• 3: = 1 + 1 + 1
• ... and so on
• 9: = 1 added to itself nine times

This group is cyclic because you can get back to any element, including the identity, by moving through these steps.

Cyclic groups are foundational in understanding more complex algebraic structures and have applications ranging from number theory to systems engineering.