In Exercises 1 through 5, compute the indicated product involving the following permutations in S6: 
σ=123456, 314562 τ=123456, 241365 μ=123456, 524316
1. τσ 2. τ2σ 3. μσ2 4. σ−2τ 5. σ−1τσ
In Exercises 6 through 9, compute the expressions shown for the permutations σ, τ and μ defined prior to Exercise 1. 6. |⟨σ⟩| 7. |⟨τ2⟩| 8. σ100 9. μ100 8 only

To solve Exercise 8, we need to compute \(\sigma^{100}\), where \(\sigma\) is the permutation \(\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 1 & 4 & 5 & 6 & 2 \end{pmatrix}\).

### Step 1: Express \(\sigma\) as a product of disjoint cycles

A permutation can be written as a product of disjoint cycles by following how each element maps under the permutation.

- Start with 1. Under \(\sigma\), \(1 \mapsto 3\), \(3 \mapsto 4\), \(4 \mapsto 5\), \(5 \mapsto 6\), \(6 \mapsto 2\), and \(2 \mapsto 1\). Therefore, the cycle representation of \(\sigma\) is:

\[
\sigma = (1\ 3\ 4\ 5\ 6\ 2)
\]

This is a 6-cycle permutation.

### Step 2: Compute \(\sigma^{100}\)

Since \(\sigma\) is a 6-cycle, the order of \(\sigma\) is 6. This means \(\sigma^6 = \text{id}\), where \(\text{id}\) is the identity permutation.

To compute \(\sigma^{100}\), notice that:

\[
100 \mod 6 = 4
\]

Thus, \(\sigma^{100} = \sigma^4\).

### Step 3: Compute \(\sigma^4\)

Now we need to compute \(\sigma^4\). We already know \(\sigma = (1\ 3\ 4\ 5\ 6\ 2)\), so we can apply the cycle four times.

- After one application of \(\sigma\): \(1 \mapsto 3\)
- After two applications of \(\sigma\): \(1 \mapsto 4\)
- After three applications of \(\sigma\): \(1 \mapsto 5\)
- After four applications of \(\sigma\): \(1 \mapsto 6\)

Thus, \(\sigma^4\) is the permutation:

\[
\sigma^4 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 1 & 2 & 3 & 4 \end{pmatrix}
\]

In cycle notation, this is:

\[
\sigma^4 = (1\ 6\ 4\ 2\ 5\ 3)
\]

### Final Answer:
\[
\sigma^{100} = (1\ 6\ 4\ 2\ 5\ 3)
\]

Let me know if you want more details on this, and here are five related questions:

1. How do you calculate the inverse of a permutation in cycle notation?
2. What is the order of a permutation, and how is it related to the cycle structure?
3. How do you compute powers of a permutation efficiently?
4. What is the significance of the identity permutation in permutation groups?
5. Can you find the cycle notation for \(\mu\)?

**Tip:** Always reduce powers of permutations by their order (modulo the length of their cycles) to simplify the calculation.

In Exercises 1 through 5, compute the indicated product involving the following permutations in S6: σ=123456, 314562 τ=123456, 241365 μ=123456, 524316 1. τσ 2. τ2σ 3. μσ2 4. σ−2τ 5. σ−1τσ In Exercises 6 through 9, compute the expressions shown for the permutations σ, τ and μ defined prior to Exercise 1. 6. |⟨σ⟩| 7. |⟨τ2⟩| 8. σ100 9. μ100

Explore how to compute the products and powers of permutations in S6. This detailed solution covers concepts like cycle notation, permutation order, and modular arithmetic. Learn how to simplify powers such as σ100 and solve related problems. Suitable for undergraduate students studying group theory.

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