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.