Let's analyze the permutation $\sigma = (24)(1759)(428)(735) \in S_9$ step by step.

1. Writing $\sigma$ as a product of disjoint cycles:

To express $\sigma$ as a product of disjoint cycles, we need to apply the cycles sequentially to the elements of the set $\{1, 2, \dots, 9\}$.

Step-by-Step Process:

• Start with element 1:

  • $1$ goes to $7$ (from the cycle $(1759)$),
  • $7$ goes to $3$ (from the cycle $(735)$),
  • $3$ goes to $5$ (from the cycle $(735)$),
  • $5$ goes back to $1$ (from the cycle $(735)$).
  • So, $1$ is part of the cycle $(1735)$.

• Next, consider element 2:

  • $2$ goes to $4$ (from the cycle $(24)$),
  • $4$ goes to $2$ (from the cycle $(428)$).
  • So, $2$ and $4$ form the cycle $(24)$.

• Consider element 8:

  • $8$ goes to $4$ (from the cycle $(428)$),
  • But $4$ already belongs to the cycle $(24)$, and 8 hasn't been involved elsewhere, meaning 8 stays fixed in $\sigma$.

• Lastly, consider element 9:

  • $9$ goes to $1$ (from the cycle $(1759)$),
  • But $1$ already belongs to the cycle $(1735)$, so 9 must return to 1, hence completing the cycle $(973)$.

Thus, the disjoint cycle form of $\sigma$ is:

$\sigma = (1735)(24)(8)$

2. Determining the order of $\sigma$:

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

• The cycle $(1735)$ has a length of 4.
• The cycle $(24)$ has a length of 2.
• The cycle $(8)$ has a length of 1.

Thus, the order of $\sigma$ is:

$\text{ord}(\sigma) = \text{LCM}(4, 2, 1) = 4$

Final Answers:

1. The disjoint cycle representation of $\sigma$ is $(1735)(24)(8)$.
2. The order of $\sigma$ is $4$.

Would you like further details on any of these steps?

Here are 5 questions related to the topic:

1. What is the significance of disjoint cycles in permutations?
2. How can you determine the order of a permutation in general?
3. Why does the cycle $(8)$ have no effect on the order of $\sigma$?
4. How would you find the inverse of $\sigma$ in its disjoint cycle form?
5. Can a permutation be written in more than one disjoint cycle form?

Tip: Remember that the identity permutation has an order of 1, and every element is mapped to itself in disjoint cycles.