How to Calculate π^σ If We Have 2 Disjoint Cycles?

[FoxMCloud]

Suppose we have 2 disjoint cycles π and σ. How can one calculate π^σ?
I know how to calculate σ^2 or σ^3 but I can't figure out how to solve that.

 

[Stephen Tashi]

Are you talking about cycles in the sense of cyclic permutations? The first question would be how we define $\pi^\sigma$.

 

[FoxMCloud]

Yes cyclic permuations. For example if$\pi$= (147)(263859) and σ=(16789)(2345) how can we calculate $\pi$^σ

 

[R136a1]

Yes cyclic permuations. For example if$\pi$= (147)(263859) and σ=(16789)(2345) how can we calculate $\pi$^σ

How did you define $\pi^\sigma$? Did you define it as $\sigma \circ \pi \circ \sigma^{-1}$?

 

[FoxMCloud]

I think it's something along the lines of σ^(-1)*$\pi$*σ. But then again, I'm not seeing how can I calculate σ^(-1).

 

[R136a1]

I think it's something along the lines of σ^(-1)*$\pi$*σ. But then again, I'm not seeing how can I calculate σ^(-1).

If you have one cycle, then you can find the inverse by reversing the cycle. So if $$\sigma = (1 ~2~5~3)$$, then $$\sigma^{-1} = (3~5~2~1)$$

Then if you have a more general form, then you can calculate the inverse by the formula $(\sigma\tau)^{-1}= \tau^{-1}\sigma^{-1}$.

For example, if you have $(1~4~6)(3~2)$, then the inverse is $(2~3)(6~4~1)$.

So now you can find $\sigma^{-1}$ and thus also $\pi^\sigma$. However, for a general theorem which makes this a LOT easier: http://www.proofwiki.org/wiki/Cycle_Decomposition_of_Conjugate