Final answer:
To prove that στσ⁻¹ is a k-cycle, we need to show that it has the same number of elements as τ, and that it cycles through those elements in the same way as τ. This can be done by applying the permutation σ, then its inverse σ⁻¹, and finally σ again to the elements of the k-cycle τ.
Step-by-step explanation:
To prove that στσ⁻¹ is a k-cycle, we need to show that it has the same number of elements as τ, and that it cycles through those elements in the same way as τ.
Let's say τ = (a₁ a₂ ... aₖ), which means τ sends a₁ to a₂, a₂ to a₃, and so on until aₖ is sent to a₁. Now, we are given σ and we want to show that στσ⁻¹ is a k-cycle.
When we apply the permutation σ, each of the elements a₁, a₂, ..., aₖ is sent to some other element in Sₙ. When we apply σ⁻¹, the elements are sent back to their original positions. Finally, when we apply σ again, the elements are sent to new positions according to σ, but in the same relative order as in τ. Thus, each element is cycled through in the same way as in τ, proving that στσ⁻¹ is a k-cycle.