Question

Consider the permutations \[ \sigma = (1 2 3 4 5 6 7 8), \quad \tau = (1 2 3 4 5 6 7 8) (4 5 3 7 8 6 1 2), \] in \( S_8 \). The number of \( \eta \in S_8 \) such that \( \eta^{-1} \sigma \eta = \tau \) is equal to ............

Hint:

In symmetric groups, conjugacy classes are determined by the cycle structure of the permutations. The number of elements that conjugate a given permutation into another is 1 if they have the same cycle type.

Answer 1

Correct Answer: -0.01 - 0.01

Step 1: Understanding the problem.
We are given two permutations \( \sigma \) and \( \tau \) in \( S_8 \). The number of elements \( \eta \in S_8 \) such that \( \eta^{-1} \sigma \eta = \tau \) is the number of elements \( \eta \) that conjugate \( \sigma \) into \( \tau \).
Step 2: Analyze conjugacy in symmetric groups.
The number of conjugates of a permutation \( \sigma \) in \( S_n \) is determined by the number of elements in its conjugacy class. Conjugacy classes in \( S_n \) are determined by the cycle type of the permutation.
Step 3: Cycle type of \( \sigma \) and \( \tau \).
Both \( \sigma \) and \( \tau \) are 8-cycles, and they have the same cycle structure. Since conjugacy preserves cycle type, there is exactly one element \( \eta \) such that \( \eta^{-1} \sigma \eta = \tau \).
Step 4: Conclusion.
Thus, the correct answer is 1.