The discussion revolves around the cycle decomposition of permutations, specifically examining the relationship between a cycle α and a permutation π, and how to express the cycle structure of the conjugate permutation παπ⁻¹.
Some participants have made attempts to compute specific cases and have shared their findings, while expressing confusion about the generalization of the results. There is an ongoing exploration of the cycle structure and how it relates to the original cycle.
Participants are working under the constraints of understanding the properties of permutations and cycles without providing complete solutions. There is a noted lack of consensus on the generalization of the results, indicating a need for further clarification.
FlickS said:Homework Statement
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)).
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.
FlickS said: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.
Okay, that definitely makes its more clear. Thanks so much!Dick said: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?