given two disjoint β and α, prove that ραρ−1 and ρβρ−1 are disjoint

Question

If $\alpha,\beta,\rho \in S_n$ with $\beta,\alpha$ disjoint then $\rho\alpha\rho -1$ and $\rho \beta \rho - 1$ are disjoint.

I know that two permutations are called disjoint if no element is moved by both, and an element by definition is moved by $\alpha$ if $\alpha(x)\neq x$.

TIA

Answer 1

Hint: Choose some $i\in \{1,\ldots,n\}$. What can you say about $\alpha(\rho^{-1}i)$ and $\beta(\rho^{-1}i)$? What does that tell you about $\rho\alpha\rho^{-1}(i)$ and $\rho\beta\rho^{-1}(i)$?