Let 𝜎 = (1 3 5 7 8)(4 3 2)(7 6)(8 2 1) be a permutation in 𝑆8
.
1. Write 𝜎 as a product of disjoint cycles.

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.

Learn how to express the permutation 𝜎 = (1 3 5 7 8)(4 3 2)(7 6)(8 2 1) as a product of disjoint cycles with detailed step-by-step analysis. Understand the concept of permutations and disjoint cycles in advanced mathematics.

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