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 need to follow the action of \(\sigma\) on each element of the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) and determine the final position of each element.

1. Start with \(1\):
   - \(1 \rightarrow 3\) (from \((1 \ 3 \ 5 \ 7 \ 8)\)),
   - \(3 \rightarrow 5\) (from \((1 \ 3 \ 5 \ 7 \ 8)\)),
   - \(5 \rightarrow 7\) (from \((1 \ 3 \ 5 \ 7 \ 8)\)),
   - \(7 \rightarrow 6\) (from \((7 \ 6)\)),
   - \(6\) maps to itself, so this ends the cycle. Thus, \(1 \rightarrow 3 \rightarrow 5 \rightarrow 7 \rightarrow 6\).

2. Next, consider the action on \(8\):
   - \(8 \rightarrow 2\) (from \((8 \ 2 \ 1)\)),
   - \(2 \rightarrow 1\) (from \((4 \ 3 \ 2)\)),
   - \(1\) has already been processed in the cycle \(1 \rightarrow 3 \rightarrow 5 \rightarrow 7 \rightarrow 6\). So \(8 \rightarrow 2 \rightarrow 1\) closes the cycle.

3. Now, look at \(4\):
   - \(4 \rightarrow 3\) (from \((4 \ 3 \ 2)\)),
   - \(3\) has already been processed, so \(4 \rightarrow 3\) is the cycle.

Finally, since \(5\) and \(7\) have already been processed, they do not form a new cycle.

Thus, the permutation \(\sigma\) can be written as a product of disjoint cycles as:
\[
\sigma = (1 \ 3 \ 5 \ 7 \ 6)(8 \ 2)
\]

This means that \(\sigma\) is the product of the two disjoint cycles \((1 \ 3 \ 5 \ 7 \ 6)\) and \((8 \ 2)\).

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Would you like further details or have any questions? Here are 5 related questions to expand your understanding:

1. How can you determine the order of a permutation from its disjoint cycles?
2. What are the cycle decompositions of the permutation \(\sigma\) in \(S_8\)?
3. How do you verify that a given permutation is a member of the symmetric group \(S_8\)?
4. Can every permutation be expressed as a product of disjoint cycles?
5. How would the expression change if the permutation were written in two-line notation?

**Tip:** When dealing with permutations, it’s helpful to track how each element is moved by the cycles, as this ensures that all cycles are properly accounted for.

Understand how to express a permutation as a product of disjoint cycles using an example. Explore the permutation 𝜎 = (1 3 5 7 8)(4 3 2)(7 6)(8 2 1) in 𝑆8, detailing each cycle's action and how to decompose it. Suitable for undergraduate-level mathematics students.

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