To express the permutation $\sigma = (1\ 3\ 5\ 7\ 8)(4\ 3\ 2)(7\ 6)(8\ 2\ 1)$ as a product of disjoint cycles, we'll trace the action of each element under $\sigma$.

Steps:

1. Start with 1:

   • $1 \rightarrow 3$ from $(1\ 3\ 5\ 7\ 8)$.
   • $3 \rightarrow 5$ from the same cycle.
   • $5 \rightarrow 7$ from the same cycle.
   • $7 \rightarrow 6$ from $(7\ 6)$.
   • $6$ is not moved by any other cycles, so $1 \rightarrow 3 \rightarrow 5 \rightarrow 7 \rightarrow 6$.

   This forms the cycle $(1\ 3\ 5\ 7\ 6)$.

2. Next, check 2:

   • $2 \rightarrow 1$ from $(8\ 2\ 1)$.
   • $1$ has already been processed, so $2$ is fixed by the remaining cycles.

   This gives the cycle $(2)$.

3. Next, check 4:

   • $4 \rightarrow 3$ from $(4\ 3\ 2)$.
   • $3$ has already been processed, so $4$ is fixed by the remaining cycles.

   This gives the cycle $(4)$.

8 has already been processed when considering the other cycles.

Final disjoint cycle form:

The permutation $\sigma$ can be written as: $\sigma = (1\ 3\ 5\ 7\ 6)(2\ 4)$

This is the product of disjoint cycles.

Would you like more details or have any questions?

Related Questions:

1. What are the cycle types of the permutation $\sigma$?
2. How do you determine the order of a permutation given in disjoint cycles?
3. How many elements in $S_8$ can be expressed as a single cycle of length 5?
4. How do you compose two permutations given in cycle notation?
5. Can every permutation be written as a product of disjoint cycles?
6. What is the inverse of the permutation $\sigma$?
7. How do disjoint cycles affect the structure of a permutation group?
8. How can you determine if a permutation is even or odd?

Tip: When working with permutations, always trace the action on each element step by step to ensure accuracy.