Problem 4 - Algebra: Pure and Applied

Question

Find all the generators of \(\mathbb{Z}_{10}, \mathbb{Z}_{12},\) and \(\mathbb{Z}_{15}\).

Step-by-Step Solution

Answer

Generators of \(\mathbb{Z}_{10}\) are \(1, 3, 7, 9\); of \(\mathbb{Z}_{12}\) are \(1, 5, 7, 11\); and of \(\mathbb{Z}_{15}\) are \(1, 2, 4, 7, 8, 11, 13, 14\).

Step 1: Understanding Generators of Cyclic Groups

In a cyclic group \(Z_n\), a generator is an element whose powers produce every element of the group. For \(Z_n\) to have a generator \(g\), \(g\) must be relatively prime to \(n\). This means that the greatest common divisor (GCD) of \(g\) and \(n\) is 1.

Step 2: Finding Generators of \(\mathbb{Z}_{10}\)

For \(Z_{10}\), the elements are \(\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \}\). We need to find the elements that have GCD equal to 1 with 10. These elements are: \(1\), \(3\), \(7\), and \(9\). Hence, these are the generators of \(Z_{10}\).

Step 3: Finding Generators of \(\mathbb{Z}_{12}\)

For \(Z_{12}\), the elements are \(\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \}\). The elements that are relatively prime to 12 are those with GCD equal to 1. These elements are: \(1\), \(5\), \(7\), and \(11\). Therefore, the generators of \(Z_{12}\) are \(1\), \(5\), \(7\), and \(11\).

Step 4: Finding Generators of \(\mathbb{Z}_{15}\)

For \(Z_{15}\), we consider the elements \(\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \}\). The elements that are coprime to 15 are \(1\), \(2\), \(4\), \(7\), \(8\), \(11\), \(13\), and \(14\). Thus, the generators of \(Z_{15}\) are \(1\), \(2\), \(4\), \(7\), \(8\), \(11\), \(13\), and \(14\).

Key Concepts

Generators

In the realm of cyclic groups, the concept of a "generator" is a cornerstone. A generator of a cyclic group is an element that, through its repeated multiplication, hits every other element within the group. Think of it as the key player who can bring every other member into the spotlight. In mathematical terms, if you have a cyclic group \(\mathbb{Z}_n\), then a generator \(g\) must be an element such that:

• \(g, g^2, g^3, \ldots, g^k\) will produce all elements of the group \(\mathbb{Z}_n\).

By using an element from the group as the base, repeatedly applying this element's "power" results in the entire group appearing. This is why some elements in a group can serve as generators and some cannot. A crucial factor is that a generator \(g\) must be relatively prime to \(n\), the order of the group. Only then can the powers of \(g\) reach all elements.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a foundational concept in mathematics. It is essential for understanding generators of cyclic groups. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. Formally, for two integers \(a\) and \(b\), the GCD is denoted as \(\text{GCD}(a, b)\). Consider the numbers 8 and 12. Their common divisors are 1, 2, and 4. Therefore, their GCD is 4 - the greatest among them.

• The GCD is crucial because a generator \(g\) for a cyclic group \(\mathbb{Z}_n\) requires that \(\text{GCD}(g, n) = 1\).

This condition ensures that \(g\) will, through its multiples, cycle through all the possible elements in the group without repeating prematurely. Hence, calculating the GCD helps to confirm whether an element can be a generator.

Relatively Prime Numbers

Numbers are considered relatively prime if their greatest common divisor (GCD) is 1. This property is pivotal for identifying generators in cyclic groups. Relatively prime numbers do not have any common factors other than 1, meaning they do not "fit" into each other in a neat multiplication table.

• For instance, 7 and 20 are relatively prime because their GCD is 1.

This relationship is key in cyclic groups like \(\mathbb{Z}_n\), where potential generators must be relatively prime to \(n\). If the GCD of \(g\) and \(n\) is anything other than 1, then there will be some elements \(g\) will miss, thus failing to "generate" the entire group. Therefore, finding elements that are relatively prime to the order of the group \(n\) is crucial for generators.