Finding permutation $a$ given $b$ and conjugate $a^b$

Question

Normally we define a conjugate relationship as

$$a^b = b~a~b^{-1}$$

But I don't know how to find $a$ given that we know $b$ and $a^b$.

Answer 1

Consider $f_b(a) = a^b = bab^{-1}$. What is the inverse function of this? If we apply the inverse of $f_b$ to $a^b$, we will get $a$.

HINT: $f_x(f_y(a))=f_{xy}(a)=xya(xy)^{-1}$. (For what $x$ is $xb$ the identity?)