Find the unknown permutation

Question

Find a permutation $x$ such that

For $$y = (1274)(356)$$
$$x^{-1}yx = (254)(1736)$$

Answer 1

It is easier if you rewrite it as $$y = (1274)(356)\\x^{-1} y x = (1736)(254),$$ so that cycles of the same length lie under each other. Then the permutation $x$ can have many choices, but one example is $$ x(1) = 1, x(7) = 2, x(3) = 7, x(6) = 4, \text{etc.} ,$$ that is, map each symbol in the line representing $x^{-1} y x$ to the corresponding symbol in the line representing $y$. This is because, for example, $x^{-1} y x$ applied to $7$ will be $3$, computed as follows: $x$ takes $7$ to $2$, $y$ takes $2$ to $7$, and $x^{-1}$ takes $7$ to $3$.