Finding cycles

Tags:
1. Mar 4, 2015

FlickS

1. The problem statement, all variables and given/known data
Let α = (α1α2...αs) be a cycle, for positive integers α1α2...αs. Let π be any permutation that παπ-1 is the cycle (π(α1)πα2...π(αs)).

2. Relevant equations

3. The attempt at a solution
I started by choosing a specific α and π, and tried finding παπ-1 to give myself some idea of what to do but have had no luck. Suggestions would be welcomed.

2. Mar 4, 2015

Dick

For example, work out what is $\pi \alpha \pi^{-1}(\pi(\alpha_1))$?

3. Mar 4, 2015

FlickS

I would get παπ−1(π(α1)) = πα(α1)) = πα2?
It gives me the next element in the cycle. So παπ−1 would be that cycle.
I'm still relatively confused.

4. Mar 4, 2015

Dick

Let's set $\sigma=\pi \alpha \pi^{-1}$ for short. You've shown $\sigma(\pi(\alpha_1))=\pi(\alpha_2)$. Generalizing that I'd say the cycle structure of $\sigma$ is $(\pi(\alpha_1)\pi(\alpha_2)...)$. Still confused?

5. Mar 4, 2015

FlickS

Okay, that definitely makes its more clear. Thanks so much!