# Finding element of order in the symmetric group

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 $$S_{12}$$
• $35$ is the best you can do with just two cycles, but once $n$ gets large enough you can do better with three separate cycles. $12$ is large enough that three cycles gives a higher LCM. Values of the maximum are given in oeis.org/A000793 – Ross Millikan Jan 1 at 15:26
Hint: $$35 = 5 \cdot 7$$ and $$5+7=12$$.