How $\beta$ and $\gamma$ are related to conjugation

Question

Let $\gamma$ and $\beta$ be permutations in $S_n$. Describe how $\gamma^{-1}\beta\gamma$ is related to $\beta$ and $\gamma$.

I dont know how to approach this question.

I know it is conjugation but how is it related to $\beta$ and $\gamma$?

Answer 1

Hint: If $\beta=(b_{i_1}\dots b_{i_{l_1}})(b_{i_2}\dots b_{i_{l_2}})\dots(b_{i_m}\dots b_{i_{l_m}})$ is the cyclic decomposition of $\beta$, then consider

$$(\gamma^{-1}(b_{i_1})\dots \gamma^{-1}(b_{i_{l_1}}))(\gamma^{-1}(b_{i_2})\dots\gamma^{-1}( b_{i_{l_2}}))\dots(\gamma^{-1}(b_{i_m})\dots\gamma^{-1}( b_{i_{l_m}})),$$

where $\gamma^{-1}(b)$ is $\gamma^{-1}$ evaluated at $b$.