Let
𝜎 = (
1 2 3 4 5 6 7 8 9
9 7 8 3 2 6 5 4 1)
be a permutation in 𝑆9
.
1. Write 𝜎 as a product of disjoint cycles

Determine the order

To write the permutation \(\sigma\) as a product of disjoint cycles and determine its order, we will follow these steps:

1. Identify the cycles by tracking where each element maps.
2. Write \(\sigma\) in cycle notation.
3. Determine the order of \(\sigma\) by finding the least common multiple (LCM) of the lengths of the disjoint cycles.

Given permutation \(\sigma\):

\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
9 & 7 & 8 & 3 & 2 & 6 & 5 & 4 & 1
\end{pmatrix}
\]

### Step 1: Identify the Cycles

Start with 1:
\[ 1 \rightarrow 9 \rightarrow 1 \]
This forms the cycle \((1 \ 9)\).

Move to the next smallest number not yet included in a cycle, which is 2:
\[ 2 \rightarrow 7 \rightarrow 5 \rightarrow 2 \]
This forms the cycle \((2 \ 7 \ 5)\).

Move to the next smallest number not yet included in a cycle, which is 3:
\[ 3 \rightarrow 8 \rightarrow 4 \rightarrow 3 \]
This forms the cycle \((3 \ 8 \ 4)\).

Number 6 maps to itself:
\[ 6 \rightarrow 6 \]
This forms the cycle \((6)\), but we typically omit single element cycles in notation.

### Step 2: Write \(\sigma\) in Cycle Notation

Combining all the disjoint cycles, we get:
\[ \sigma = (1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4) \]

### Step 3: Determine the Order of \(\sigma\)

The order of a permutation is the LCM of the lengths of its disjoint cycles.

The lengths of the cycles are:
- \((1 \ 9)\): length 2
- \((2 \ 7 \ 5)\): length 3
- \((3 \ 8 \ 4)\): length 3

Calculate the LCM of 2, 3, and 3:
\[ \text{LCM}(2, 3, 3) = 6 \]

Therefore, the order of \(\sigma\) is 6.

### Summary

- The permutation \(\sigma\) can be written as the product of disjoint cycles: \((1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4)\).
- The order of \(\sigma\) is 6.

Do you want more details or have any questions? Here are some related questions to expand your understanding:

1. How do you find the cycle structure of a permutation in \(S_n\)?
2. What is the significance of disjoint cycles in permutation groups?
3. How do you calculate the least common multiple (LCM) of multiple numbers?
4. What are the properties of the order of a permutation?
5. How do single element cycles affect the cycle notation?
6. What is the symmetric group \(S_n\) and its properties?
7. How do permutations relate to matrix representations?
8. How can you determine if two permutations are conjugate in \(S_n\)?

**Tip:** When working with permutations, always ensure that you check the mapping of each element carefully to avoid mistakes in identifying cycles.

Learn how to express permutation \(\sigma\) as a product of disjoint cycles and calculate its order using the least common multiple (LCM). This problem involves understanding permutation structures and cycle notation in permutation groups.

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