Question

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

Answer 1

Correct Answer: 6

To solve the problem, we need to determine the number of elements in the centralizer of the permutation σ in the symmetric group S₅. The permutation σ is given by the cycle notation as σ = (1 2 3)(4 5). The centralizer N(σ) consists of all permutations τ in S₅ that commute with σ, meaning σ ∘ τ = τ ∘ σ. 

Since σ can be decomposed into disjoint cycles, we can analyze its centralizer by considering the centralizers of each cycle separately. For the cycle (1 2 3):
- It has 3 elements (1, 2, 3), so its centralizer consists of all permutations that permute these three elements in a way that maintains the cycle structure. This forms a subgroup of S₃, generated by (1 2 3) and (1 3 2), which is a cyclic group of order 3.

For the cycle (4 5):
- It has 2 elements, meaning its centralizer consists of all permutations of 4 and 5 that preserve the cycle structure. This forms S₂, which has an order of 2.

The total number of elements in N(σ) is the product of centralizers' sizes for these disjoint cycles. Therefore, |N(σ)| = 3 × 2 = 6.