Find element of Group

Question

Let$ (1 3 5 7 )$ and $(2 3 6 8)$ be elements of $S_8$. Find a element$π$ form $S_8$ for which it is worth $π (1 3 5 7 ) π^{-1} = (2 3 6 8)$

Answer 1

We know that if $π\in S_n$ and $(a_1, a_2,...,a_n)\in S_n $ Then $$ π(a_1, a_2,...,a_n)π^{-1}=(π(a_1),π(a_2),...,π(a_n))$$

Now from your question we get $π(1)=2,π(3)=3,π(5)=6,π(7)=8$ Image of $π$ for other elements we can choose at many ,like$π(2)=5,π(4)=1,π(6)=7,π(8)=4$ Then we get $ π=(1,2,5,6,7,8,4)$,

You can choose those four images (but for that choosing $π$ must be a bijective mapping) another way and get another$π$

Answer 2

For instance, $(12)(56)(78)$. There are others.