To compute the number of elements in the conjugacy class of the permutation $\sigma$ in $S_7$, we need to use the formula related to the cycle structure of $\sigma$:

Step 1: Identify the cycle structure of $\sigma$

Given the permutation (\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \ 5 & 4 & 2 & 3 & 6 & 7 & 1 \end{pmatrix}), we previously determined that the cycle structure is:

$\sigma = (1 \, 5 \, 6 \, 7)(2 \, 4 \, 3)$

This means:

• There is one cycle of length 4 ($(1 \, 5 \, 6 \, 7)$).
• There is one cycle of length 3 ($(2 \, 4 \, 3)$).

Step 2: Apply the conjugacy class size formula

The size of the conjugacy class of a permutation $\sigma$ in $S_n$ is given by:

$|C(\sigma)| = \frac{n!}{z_1^{m_1} m_1! \times z_2^{m_2} m_2! \times \dots \times z_k^{m_k} m_k!}$

Where:

• $n$ is the total number of elements (here $n = 7$).
• $z_i$ is the length of the $i$-th cycle.
• $m_i$ is the number of cycles of that length.

For $\sigma$:

• $z_1 = 4$, $m_1 = 1$ (one cycle of length 4).
• $z_2 = 3$, $m_2 = 1$ (one cycle of length 3).

Step 3: Plug in the values

Using the values for $n = 7$, $z_1 = 4$, $m_1 = 1$, $z_2 = 3$, and $m_2 = 1$:

$|C(\sigma)| = \frac{7!}{4^1 \cdot 1! \times 3^1 \cdot 1!} = \frac{5040}{4 \times 3} = \frac{5040}{12} = 420$

Conclusion

The number of elements in the conjugacy class of $\sigma$ is $|C(\sigma)| = 420$.

Would you like any more details or have any questions?

Related Questions:

1. How does the cycle structure of a permutation determine its conjugacy class in a symmetric group?
2. Why do conjugacy classes in symmetric groups depend on cycle lengths rather than specific elements?
3. What is the significance of the factorial terms in the formula for the size of a conjugacy class?
4. How would the size of the conjugacy class change if $\sigma$ had multiple cycles of the same length?
5. Can two permutations in the same symmetric group have the same conjugacy class size but different cycle structures?

Tip:

When calculating the size of a conjugacy class, make sure to clearly identify the cycle structure first, as this determines the factorial and power terms in the formula.