Let σ =

1 2 3 4 5 6 7
5 4 2 3 6 7 1 
∈ S7

Compute the number of elements in the conjugacy class of σ

To compute the number of elements in the conjugacy class of a permutation \( \sigma \) in the symmetric group \( S_7 \), we need to consider the cycle structure of \( \sigma \). The number of elements in the conjugacy class of \( \sigma \) is given by:

\[
\frac{n!}{z_1^{m_1} m_1! \cdot z_2^{m_2} m_2! \cdots z_k^{m_k} m_k!}
\]

where:
- \( n = 7 \) is the number of elements being permuted.
- \( z_i \) is the length of the \( i \)-th cycle.
- \( m_i \) is the number of cycles of length \( z_i \) in \( \sigma \).

### Step 1: Determine the cycle structure of \( \sigma \).

Given:
\[
\sigma = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
5 & 4 & 2 & 3 & 6 & 7 & 1
\end{pmatrix}
\]

We write \( \sigma \) in its cycle notation:

- \( 1 \) maps to \( 5 \), and \( 5 \) maps to \( 6 \), and \( 6 \) maps to \( 7 \), and \( 7 \) maps back to \( 1 \). So, we have the cycle \( (1 \ 5 \ 6 \ 7) \).
- \( 2 \) maps to \( 4 \), and \( 4 \) maps back to \( 3 \), and \( 3 \) maps to \( 2 \). So, we have the cycle \( (2 \ 4 \ 3) \).

Thus, \( \sigma \) can be written as:
\[
\sigma = (1 \ 5 \ 6 \ 7)(2 \ 4 \ 3)
\]

### Step 2: Identify the cycle lengths.

The cycle lengths in \( \sigma \) are:
- One cycle of length 4: \( (1 \ 5 \ 6 \ 7) \)
- One cycle of length 3: \( (2 \ 4 \ 3) \)

So, \( z_1 = 4 \), \( m_1 = 1 \), \( z_2 = 3 \), and \( m_2 = 1 \).

### Step 3: Calculate the size of the conjugacy class.

The number of elements in the conjugacy class of \( \sigma \) is:
\[
\frac{7!}{4^1 \cdot 1! \cdot 3^1 \cdot 1!} = \frac{5040}{4 \times 3} = \frac{5040}{12} = 420
\]

Thus, the number of elements in the conjugacy class of \( \sigma \) is **420**.

Would you like more details or have any questions?

Here are five related questions:
1. How do you compute the number of elements in a conjugacy class for a permutation in \( S_n \)?
2. What is the significance of cycle notation in understanding permutations?
3. How can one determine the cycle structure of a given permutation?
4. What role do the lengths of cycles play in determining conjugacy classes?
5. How do the concepts of cycle type and cycle structure relate to the symmetric group?

**Tip:** Understanding the cycle structure of permutations is key to many aspects of group theory, particularly in analyzing the symmetric groups and their properties.

Learn how to compute the number of elements in the conjugacy class of a permutation σ in the symmetric group S7. Understand the cycle structure of σ and apply the formula n! / (z1^m1 * m1! * z2^m2 * m2!) to determine the size of the conjugacy class. Suitable for advanced undergraduate level studies.

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