So I have to find all $x \in S_5$ such that $x^3 = (12)$. For example, one solution would be $(12)$ itself, because its order is $2$. How can I find all of the solutions though? Is it just trial and error?
(also, the multiplication is defined from left to right, just fyi)
$o(x)$ is a divisor of $6$ means $o(x)=1,2,3$ 0r $6$. $o(x)=1,3$ are automatically excluded. For $o(x)=2$, $x=(12)$ is the only solution. For $x=6$, $x=(12)(abc)$ is a solution, where $a,b,c\in \{3,4,5\}$.