I'm just starting my study of group theory, and I'm trying to figure out how to find every element in some order of a given group. Say I have the group $S_4$ (symmetric group of degree $4$). How could I find, say, all of the elements of order $3$, or $4$, or $5$? Is there some general process to find all elements for an order $n$?
The elements of order $k$ in $S_n$ have cycles of lengths with lcm $k$.
For example: Elements of order 6 in $S_4$ would have to have either a 6-cycle, but there are no 6 distinct elements or at least an even cycle and a cycle whose length is divisible by three, but $2+3 > 4$, so this isn't possible, either. Therefore, there are no elements of order 6 in $S_4$
The elements of order 3 in $S_4$ are the 3-cycles, so there are $8$ such elements.
For groups whose order is divisible by a prime $p$ but not by $p^2$, the restriction on numbers of $p$-Sylow groups can be used to calculate the number of elements of order $p$.