What is the proper and technical way to find if there exists an element in the symmetric group $S_n$ (Wikipedia article on symmetric groups.). I do know how to disprove if there does not exists such elements, but how should I find the element which does exists of this order?
For example, in the $S_{12}$, there is not such element of order $13$ because The only elements of order $13$ in $S_n$ are unions of disjoint $13$-cycles, since $13$ is prime. This would require $S_{12}$ to contain at least $13$ symbols, which it does not. But I think that there is an element of order $35$ in $S_{12}$. How should I find it?
EDIT: I do understand that I have to find a $7$-cycle and a $5$-cycle, but how?
If you want an element of order $k$, you can factor $k$ and make some cycles whose $\operatorname {LCM}$ is $k$. In your example of $35$ we note that $35=5\cdot 7$, so if you make a $5-$cycle and a $7-$cycle you will have an element of order $35$. As $5+7=12$ you have enough room to do this in $S_{12}$