I'm looking for a permutation $\sigma\in S_{4}$ so $\sigma^{2}=(1,2)(3,4)$.
I know that $\sigma=(1,3,2,4)$ will solve it but how formally prove it?
I came across with another one: Finding $\sigma\in S_{5}$ so $\sigma^{2}=(1,2,3,4,5)$.
That one was easy because if $\alpha=(1,2,3,4,5)$ then $\alpha^{5}=id$ so $(\alpha^{3})^{2}=\alpha$ meaning $\alpha^{3}$ solves the equation. we get: $\sigma=\alpha^{3}=(1,4,2,5,3)$.
How similarity to solve the first question?
Max order of element in $S_4$ is 4. The order of $\sigma$ is greater than $2$. But it can't be $3$, as those permutations always fix one element, and $\sigma^2$ has all elements moved. So, it is some cycle of length 4: $(1\star\star\,\star)$, next element can't be 2. Let's say it is 3, then 3 should be moved to 2: $\sigma_1 = (1324)$. Since the order of this cycle is $2\times2=4$, then $\sigma_2=\sigma_1^{-1} = (1423)$ also works fine.