I am realizing that I am not sure I quite understand permutations, especially in this context. The specific problem I am working with asks to find the smallest possible $n$ such that $S_{n}$ has an element of order 6. I am confused and was wondering what steps I should take. Upon reading other similar posts, I know that I need to find the prime factorization of 6, and the least common multiple of something but I am not quite sure what this "something" is.
Any hints/help are appreciated, and I can revise my question to include an attempt once I understand these definitions/basics a little more.
First, let $n$ be a positive integer, $\pi$ be a permutation in $S_n$, and let $r_1,\ldots, r_k$ be the lengths of all the cycles in $\pi$ [the $r_j$s are all positive integers]. Then on the one hand: $$\sum_{i=1}^k r_j = n,$$ and on the other hand, $$|\pi| = \text{lcm}(r_1,\ldots, r_k),$$ or equivalently, $|\pi|$ is the lcm of the cycle lengths of $\pi$. This is what you need to solve this exercise.
So suppose there is a permutation $\pi$ in $S_4$ where $|\pi|=6$. Then the $\text{lcm}$ of the cycle lengths of $\pi$ has to be $6$ and the sum of the cycle lengths has to be $4$. It is easy to check, however, that there are no set of positive integers that sum to $4$ and have an lcm of $6$ though, so this is impossible. Thus, for $n=4$, there is no $\pi \in S_4$ where $|\pi|$ is $6$, after all. So $n$ must satisfy $n >4$.
On the other hand, for $n=5$, note that $\pi \in S_5$ specified $\pi$ $=$ $(123)(45)$ indeed satisfies $|\pi| =6$. So $n \le 5$, and thus from the previous paragraph, $n=5$ is the smallest such integer. ■