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Chapter 4: Problem 5

Find the order of every element in \(\mathbb{Z}_{18}\).

Short Answer

Expert verified
The orders of the elements in \(\mathbb{Z}_{18}\) are: - Order of \(0\) is Undefined - Order of \(1\) is \(1\) - Order of \(2\) is \(9\) - Order of \(3\) is \(6\) - Order of \(4\) is \(3\) - Order of \(5\) is \(9\) - Order of \(6\) is \(2\) - Order of \(7\) is \(9\) - Order of \(8\) is \(3\) - Order of \(9\) is \(2\) - Order of \(10\) is \(3\) - Order of \(11\) is \(9\) - Order of \(12\) is \(2\) - Order of \(13\) is \(9\) - Order of \(14\) is \(3\) - Order of \(15\) is \(6\) - Order of \(16\) is \(9\) - Order of \(17\) is \(9\)

Step by step solution

01

List the elements of \(\mathbb{Z}_{18}\)

The set \(\mathbb{Z}_{18}\) contains the integers from \(0\) to \(17\), which can be expressed as \(\{0, 1, 2, 3, ..., 17\}\).
02

Compute the orders of each element

We will now compute the order of each element \(a\) by finding the smallest positive integer \(n\) that satisfies \(a^n \equiv 1 \pmod{18}\). 1. The order of \(0\) is undefined because \(0^n\) is always \(0\) and never \(1\). 2. The order of \(1\) is \(1\) because \(1^1 \equiv 1 \pmod{18}\). 3. The order of \(2\) is \(9\) because \(2^9 \equiv 1 \pmod{18}\). 4. The order of \(3\) is \(6\) because \(3^6 \equiv 1 \pmod{18}\). 5. The order of \(4\) is \(3\) because \(4^3 \equiv 1 \pmod{18}\). 6. The order of \(5\) is \(9\) because \(5^9 \equiv 1 \pmod{18}\). 7. The order of \(6\) is \(2\) because \(6^2 \equiv 1 \pmod{18}\). 8. The order of \(7\) is \(9\) because \(7^9 \equiv 1 \pmod{18}\). 9. The order of \(8\) is \(3\) because \(8^3 \equiv 1 \pmod{18}\). 10. The order of \(9\) is \(2\) because \(9^2 \equiv 1 \pmod{18}\). 11. The order of \(10\) is \(3\) because \(10^3 \equiv 1 \pmod{18}\). 12. The order of \(11\) is \(9\) because \(11^9 \equiv 1 \pmod{18}\). 13. The order of \(12\) is \(2\) because \(12^2 \equiv 1 \pmod{18}\). 14. The order of \(13\) is \(9\) because \(13^9 \equiv 1 \pmod{18}\). 15. The order of \(14\) is \(3\) because \(14^3 \equiv 1 \pmod{18}\). 16. The order of \(15\) is \(6\) because \(15^6 \equiv 1 \pmod{18}\). 17. The order of \(16\) is \(9\) because \(16^9 \equiv 1 \pmod{18}\).
03

Present the orders of the elements

Now we can present the orders of the elements in \(\mathbb{Z}_{18}\): Element: Order 0: Undefined 1: 1 2: 9 3: 6 4: 3 5: 9 6: 2 7: 9 8: 3 9: 2 10: 3 11: 9 12: 2 13: 9 14: 3 15: 6 16: 9 17: 9

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Order of an Element
The order of an element in a group is a central concept in group theory. It helps us understand how elements behave when repeated operations are applied.
When we talk about the 'order of an element,' we're looking for the smallest positive integer, say \(n\), such that the repeated operation of the element \(a\) results in the identity element of the group. In a group like the integers modulo 18, denoted as \(\mathbb{Z}_{18}\), this operation is addition.
For example, in \(\mathbb{Z}_{18}\), the order of an element \(a\) is the smallest positive integer \(n\) for which \(a \times n \equiv 0 \pmod{18}\). If \(a^n \equiv 1 \pmod{18}\) for multiplication, then \(a\) has order \(n\). When we calculate this for each element, we learn more about the structure of \(\mathbb{Z}_{18}\).
Some elements repeat themselves quickly back to the identity, having a small order, while others take more repetitions.
Understanding the order of elements is essential for breaking down and analyzing cyclic groups and their properties.
Modular Arithmetic
Modular arithmetic is like clock math鈥攊t repeats itself after a set number of steps. When figuring out equations, each result is always brought back within a range, called modulus. This mathematical tool is critical for solving problems like finding the order of elements in sets such as \(\mathbb{Z}_{18}\).
To understand modular arithmetic, consider how a clock works:
  • Hours repeat every 12 hours; minutes repeat every 60 minutes.
  • Adding 10 hours to 9 o'clock moves us to 7 o'clock, not 19 o'clock, because after 12, we start from 0 again.
  • In modulo 18, numbers from 0 to 17 are used. Any calculation wraps back into this range.
It's used to define operations in groups, especially in cases like \(\mathbb{Z}_{18}\), helping solve congruence relations like \(a^n \equiv 1 \pmod{18}\).
This art of wrapping numbers helps keep equations simple, making it a foundational element in group theory and coding systems.
Group Theory
Group theory is a branch of mathematics exploring sets combined with an operation that meets specific criteria. It's these properties that make a set of elements behave like a group:
  • Closure: Performing the operation on any elements of the set results in a member of the set.
  • Associativity: The operation's grouping doesn't change the result.
  • Identity: There exists an element that leaves others unchanged when used in the operation.
  • Inverse: Each element has a counterpart, bringing it back to the identity element.
In cyclic groups like \(\mathbb{Z}_{18}\), the entire group can be generated using one element through repeated operations.
Group theory ties into fields such as number theory, cryptography, and algebra. To find the order of an element, which means knowing how often an element needs combining with itself to recreate the identity element, is a group theory exercise.
Understanding group properties is crucial for recognizing patterns, predicting behaviors, and solving problems across various mathematical and practical applications.

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Prove that the circle group is a subgroup of \(\mathbb{C}^{*}\).

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Find all elements of finite order in each of the following groups. Here the "*" indicates the set with zero removed. (a) \(\mathbb{Z}\) (b) \(\mathbb{Q}^{*}\) (c) \(\mathbb{R}^{*}\)
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