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

What is the order and sgn

To solve the given permutation problem, we will:

1. Write the permutation \(\sigma\) as a product of disjoint cycles.
2. Determine the order of the permutation.
3. Determine the sign (sgn) of the permutation.

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: Write \(\sigma\) as a Product of Disjoint Cycles

Identify the cycles by following the mapping of each element:

- 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.

Combining all the disjoint cycles, we get:

\[
\sigma = (1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4)
\]

### Step 2: Determine the Order of \(\sigma\)

The order of a permutation is the least common multiple (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.

### Step 3: Determine the Sign (sgn) of \(\sigma\)

The sign of a permutation is \((-1)^k\), where \(k\) is the number of transpositions in the cycle decomposition.

Count the transpositions in the cycle notation:
- \((1 \ 9)\) is 1 transposition.
- \((2 \ 7 \ 5)\) is 2 transpositions: \((2 \ 7)\) and \((7 \ 5)\).
- \((3 \ 8 \ 4)\) is 2 transpositions: \((3 \ 8)\) and \((8 \ 4)\).

So, the total number of transpositions \(k\) is:

\[
1 + 2 + 2 = 5
\]

The sign (sgn) of the permutation is:

\[
(-1)^5 = -1
\]

### Summary

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

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 analyzing the sign of a permutation, remember that each cycle of length \(n\) contributes \(n-1\) transpositions. This is helpful for quickly determining the number of transpositions without explicitly writing each one out.

Learn how to analyze permutations using disjoint cycles and determine their sign (-1)^k. This problem involves a detailed step-by-step solution for a permutation in S9, including identifying disjoint cycles, calculating the order, and determining the sign. Suitable for undergraduate students and those interested in group theory.

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