Find $\pi \in S_9$ such that $\pi \circ (4,5,6,8) = (1,7,6,8,9)(2,3)$.

Question

$\pi \in S_9 \quad \pi \circ (4,5,6,8) = (1,7,6,8,9)(2,3)$

Question: Where is my mistake?

$(4,5,6,8)^{-1}=(8,6,5,4)$

$(8,6,5,1) \circ (1,7,6,8,9)(2,3)=(1,7,5,4,8,9)(2,3)$

$(1,7,5,4,8,9)(2,3) \circ (4,5,6,8) = (4)(5,6,9,1,7)(2,3) \neq (1,7,5,4,8,9)(2,3)$

Note: Because there are two ways of multipilication - I always multiply the right permutation with the left one.

Answer 1

You need to multiply by $(4\,5\,6\,8)^{-1}$ on the right in order to make $\pi\circ(4\,5\,6\,8)$ into just $\pi$.

When you multiply on the left what you get is $$ (4\,5\,6\,8)^{-1} \circ \pi \circ (4\,5\,6\,8) = (1\,7\,5\,4\,8\,9)(2\,3) $$ which does not directly tell you what $\pi$ is.