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
and find the order
A

To find the order of the permutation \( \sigma \), we first need to express \( \sigma \) as a product of disjoint cycles, which we already did in the previous step:

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

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

- The cycle \( (1 \ 9) \) has a length of 2.
- The cycle \( (2 \ 7 \ 5) \) has a length of 3.
- The cycle \( (3 \ 8 \ 4) \) has a length of 3.

Now, we compute the LCM of these lengths:

\[
\text{LCM}(2, 3, 3) = 6
\]

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

Would you like any more details or have any questions?

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**Related Questions:**

1. What is the meaning of the order of a permutation?
2. How does the order of a permutation relate to its cycles?
3. What is the order of a permutation consisting of two cycles of length 4 and 5?
4. How do you find the LCM of several numbers?
5. Can a permutation have an infinite order?
6. How do you verify the order of a permutation by applying it repeatedly?
7. What is the order of the identity permutation in \( S_9 \)?
8. How is the order of a permutation related to group theory?

**Tip:** The order of a permutation gives the number of times you must apply the permutation to return to the starting configuration.

Learn how to find the order of the permutation σ = (1 9)(2 7 5)(3 8 4) using disjoint cycles and calculating the least common multiple (LCM) of their lengths. This problem involves advanced concepts in permutations and is suitable for advanced high school students and beyond.

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