Cycle Decomposition of Permutations

[FlickS]

Homework Statement

Let α = (α₁α₂...α_s) be a cycle, for positive integers α₁α₂...α_s. Let π be any permutation that παπ^-1 is the cycle (π(α₁)πα₂...π(α_s)).

Homework Equations

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.

 

[Dick]

Homework Statement

Let α = (α₁α₂...α_s) be a cycle, for positive integers α₁α₂...α_s. Let π be any permutation that παπ^-1 is the cycle (π(α₁)πα₂...π(α_s)).

Homework Equations

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.

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

 

[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.

 

[Dick]

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.

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?

 

[FlickS]

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?

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