Let σ = (24)(1759)(428)(735) ∈ S9.
1. Write σ as a product of disjoint cycles.
2. Determine ord(σ).

Let's analyze the permutation \( \sigma = (24)(1759)(428)(735) \in S_9 \) step by step.

### 1. Writing \( \sigma \) as a product of disjoint cycles:

To express \( \sigma \) as a product of disjoint cycles, we need to apply the cycles sequentially to the elements of the set \(\{1, 2, \dots, 9\}\).

**Step-by-Step Process:**

- Start with element 1:
  - \( 1 \) goes to \( 7 \) (from the cycle \( (1759) \)),
  - \( 7 \) goes to \( 3 \) (from the cycle \( (735) \)),
  - \( 3 \) goes to \( 5 \) (from the cycle \( (735) \)),
  - \( 5 \) goes back to \( 1 \) (from the cycle \( (735) \)).
  - So, \( 1 \) is part of the cycle \( (1735) \).

- Next, consider element 2:
  - \( 2 \) goes to \( 4 \) (from the cycle \( (24) \)),
  - \( 4 \) goes to \( 2 \) (from the cycle \( (428) \)).
  - So, \( 2 \) and \( 4 \) form the cycle \( (24) \).

- Consider element 8:
  - \( 8 \) goes to \( 4 \) (from the cycle \( (428) \)),
  - But \( 4 \) already belongs to the cycle \( (24) \), and 8 hasn't been involved elsewhere, meaning 8 stays fixed in \( \sigma \).

- Lastly, consider element 9:
  - \( 9 \) goes to \( 1 \) (from the cycle \( (1759) \)),
  - But \( 1 \) already belongs to the cycle \( (1735) \), so 9 must return to 1, hence completing the cycle \( (973) \).

Thus, the disjoint cycle form of \( \sigma \) is:

\[
\sigma = (1735)(24)(8)
\]

### 2. Determining the order of \( \sigma \):

The order of a permutation is the least common multiple (LCM) of the lengths of its disjoint cycles.

- The cycle \( (1735) \) has a length of 4.
- The cycle \( (24) \) has a length of 2.
- The cycle \( (8) \) has a length of 1.

Thus, the order of \( \sigma \) is:

\[
\text{ord}(\sigma) = \text{LCM}(4, 2, 1) = 4
\]

### Final Answers:

1. The disjoint cycle representation of \( \sigma \) is \( (1735)(24)(8) \).
2. The order of \( \sigma \) is \( 4 \).

Would you like further details on any of these steps?

Here are 5 questions related to the topic:

1. What is the significance of disjoint cycles in permutations?
2. How can you determine the order of a permutation in general?
3. Why does the cycle \( (8) \) have no effect on the order of \( \sigma \)?
4. How would you find the inverse of \( \sigma \) in its disjoint cycle form?
5. Can a permutation be written in more than one disjoint cycle form?

**Tip:** Remember that the identity permutation has an order of 1, and every element is mapped to itself in disjoint cycles.

Explore how to analyze a permutation using disjoint cycles and calculate its order. Understand the significance of disjoint cycles in permutations and learn to determine the order of a permutation step-by-step. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation