Question

Let \( g \) be an element of \( S_7 \) such that \(g\) commutes with the element \((2,6,4,3)\). The number of such \(g\) is

• A. 6
• B. 4
• C. 24
• D. 48

Hint:

The size of a permutation’s centralizer in \(S_n\) depends only on its disjoint cycle structure. Use \( |C_{S_n}(\sigma)| = \prod_i k_i^{m_i} m_i! \) for cycles of length \(k_i\) repeated \(m_i\) times.

Answer 1

The Correct Option is C

Step 1: Structure of the permutation.
\((2,6,4,3)\) is a 4-cycle acting on \(\{2,3,4,6\}\). Its centralizer in \(S_7\) consists of all permutations that preserve this cycle structure.
Step 2: Compute size of centralizer.
For a \(k\)-cycle in \(S_n\), \[ |C_{S_n}(\sigma)| = k \cdot (n-k)!. \] Here, \(k=4\), \(n=7\), so \[ |C_{S_7}(\sigma)| = 4 \times 3! = 24. \]
Step 3: Conclusion.
Hence, there are 24 elements commuting with \((2,6,4,3)\).