Detecting duplicates of order 2 in $A_5$

Question

How can I intuitively think about why a certain pair of disjoint cycles is a duplicate of another? For example $(12)(34)$ is a duplicate of $(12)(43)$. Is it just a matter of brute forcing all elements of order two and checking which ones are duplicates of each other?

Answer 1

In order to count number of elements of order $2$, first you have to be know what should be look of the elements. They should be of even order It can be easily seen elements of order two in $S_n$ have look like $(12)$ and $(12)(34)$, but elements of $(12)$ are not possible. Therefore all the elements of $S_n$ which are of type $(12)(34)$ are the only elements of order $2$ in $A_5$.

$$\text{elements of same cocyle}=\frac{120}{2^2 2!}=\frac{120}{8}=15$$ Therefore total $15$ elements of order $2$ in $A_5.$ For the used formula please see the link Counting cycle structures in $S_n$.