Conjugate permutation and "their" $\alpha$

Question

Let $\sigma=(13624)(587)(9)$,$\tau =(15862)(394)(7)$.

Determine such $\alpha$ that $\alpha \sigma \alpha^{-1} = \tau $.

The elements $\sigma, \tau $ must be conjugate. But how many such $\alpha$ are there? Intuition suggests to me, exactly one.

Answer 1

You have to associate the elements of a cycle one to another. To make sure that the cycles are conjugated, you can just choose the image of one element. Then the other ones are fixed.

For instance, when you want to conjugate $(123) $ with $(456)$ you have the following possibilities:

1. $\alpha(1) = 4, \alpha(2) = 5, \alpha(3) = 6$
2. $\alpha(1) = 5, \alpha(2) = 6, \alpha(3) = 4$
3. $\alpha(1) = 6, \alpha(2) = 4, \alpha(3) = 5$

Then you have $3\times 5 = 15$ possible choices.