How to solve equation $\tau(123)(45)(67)\tau^{-1}=(765)(43)(21)$ and alike? Let $\sigma\in S_7$ be $(123)(45)(67)$. And Find $\tau\in S_7$ such that $\tau\sigma\tau^{-1}=(765)(43)(21)$. I understand that in symmetric group conjugate elements have the same cycle structure. Hence, $\tau$ should share the structure of $\sigma$. Then I guess I could trial and error to find the solution for $\tau$. However, it is not very nice to do so. I am wondering whether there is a good way to solve such equation in general case. Thank you!

Conjugation permutes elements "in place" in their cycle structure.  Hence you want $\tau$ to map 1 to 7, 2 to 6, 3 to 5, 4 to 4, 5 to 3, 6 to 2, and 7 to 1.  That is, $$\tau=(17)(26)(35)$$
