Find the solution x in S5 to the following equation

Question

I am asked to find the solution x in $S_5$to the following equation:

$(1 \ 3) \circ x \circ (2 \ 3 \ 5) = (3 \ 4)$

Here is my attempt,

$(1 \ 3)^{-1} \circ (1 \ 3) \circ x \circ (2 \ 3 \ 5) = (1 \ 3)^{-1} \circ (3 \ 4)$

$\Rightarrow \space$ $x \circ (2 \ 3 \ 5) = (3 \ 1) \circ (3 \ 4)$

$\Rightarrow \space$ $x \circ (2 \ 3 \ 5) \circ (2 \ 3 \ 5)^{-1} = (3 \ 1) \circ (3 \ 4) \circ (2 \ 3 \ 5)^{-1}$

$\Rightarrow \space$ $x = (3 \ 1) \circ (3 \ 4) \circ (5 \ 3 \ 2)$

$\Rightarrow \space$ $x = (3 \ 1) \circ (3 \ 4) \circ (5 \ 2) \circ (5 \ 3) $

Is this correct so far? Can I simplify further? Thank you for your help and time!

Answer 1

Your derivation is correct but you can go one step further and write: $$x=(1\ 3\ 2\ 5\ 4),$$ assuming that $fg(i)=f(g(i))$. For example for $3$ we have $3\to5$ because of $(5\ 3)$ and then $5\to2$ because of $(5\ 2)$, so $3\to2$.