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

To express the permutation \(\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 7 & 8 & 3 & 2 & 6 & 5 & 4 & 1 \end{pmatrix}\) as a product of disjoint cycles, we proceed as follows:

### Step-by-Step Process:

1. **Start with 1**:
   - \(\sigma(1) = 9\)
   - \(\sigma(9) = 1\)
   - So, the cycle is \((1 \ 9)\).

2. **Move to the smallest number not yet included in any cycle (2)**:
   - \(\sigma(2) = 7\)
   - \(\sigma(7) = 5\)
   - \(\sigma(5) = 2\)
   - So, the cycle is \((2 \ 7 \ 5)\).

3. **Next, take the smallest number not yet included in any cycle (3)**:
   - \(\sigma(3) = 8\)
   - \(\sigma(8) = 4\)
   - \(\sigma(4) = 3\)
   - So, the cycle is \((3 \ 8 \ 4)\).

4. **Finally, consider the number 6**:
   - \(\sigma(6) = 6\)
   - This is a fixed point and can be written as \((6)\).

### Final Answer:

Putting all the cycles together, we get:

\[
\sigma = (1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4)(6)
\]

Since \((6)\) is a fixed point, it is often omitted, so the permutation \(\sigma\) as a product of disjoint cycles is typically written as:

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

This representation shows \(\sigma\) as a product of disjoint cycles.

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Would you like more details on this solution or have any other questions?

Here are some related questions:

1. What is the order of the permutation \(\sigma\)?
2. How do you find the inverse of a permutation written as a product of disjoint cycles?
3. What is the cycle type of the permutation \(\sigma\)?
4. Can \(\sigma\) be expressed as a product of transpositions?
5. How many elements are fixed by the permutation \(\sigma\)?
6. If \(\tau = (1 \ 2 \ 3)(4 \ 5 \ 6)\), what is \(\sigma \circ \tau\)?
7. How do you compute the power of a permutation?
8. What is the sign of the permutation \(\sigma\)?

**Tip:** When working with permutations, remember that disjoint cycles commute, meaning the order in which you write them doesn't affect the overall permutation.

Learn how to express the permutation 𝜎 = (1 2 3 4 5 6 7 8 9 / 9 7 8 3 2 6 5 4 1) as a product of disjoint cycles with a step-by-step solution. Understand the concepts of permutations and disjoint cycles. Suitable for undergraduate students studying permutation theory.

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