Question

Let \( \sigma \in S_4 \) be the permutation defined by \( \sigma(1) = 2 \), \( \sigma(2) = 3 \), \( \sigma(3) = 1 \), and \( \sigma(4) = 4 \). The number of elements in the set \[ \{ \tau \in S_4 : \tau \circ \sigma^{-1} = \sigma \} \] is equal to ...............

Hint:

For problems involving permutations, carefully analyze the cycle structure of the permutation and use it to count valid solutions.

Answer 1

Step 1: Interpret the problem.
We are asked to find the number of elements \( \tau \in S_4 \) such that \( \tau \circ \sigma^{-1} = \sigma \), which is equivalent to: \[ \tau = \sigma \circ \sigma. \] Thus, we need to find the number of permutations \( \tau \) such that \( \tau \) satisfies this condition. Step 2: Calculate the number of valid \( \tau \).
By analyzing the structure of \( \sigma \), we find that there are 2 valid permutations \( \tau \) that satisfy the given equation. Final Answer: \[ \boxed{2}. \]