TITLE: Orders of a symmetric group
QUESTION [10 upvotes]: Consider the symmetric group $S_5$. I would like to find how many elements of $S_5$ are of order 5, and how many are of order 6. I would also like to determine what the maximum order of an element in this group would be.
Here is what I have so far: elements of order 5 are the 5-cycles. Elements of order 6 (since a 6-cycle is impossible for 5 elements) must have at least an even cycle and a cycle of length divisible by 3, and 2 + 3 = 5, so elements of order 5 are a combination of 2-cycles and 3-cycles.
How can I find the total count of such elements of order 5 and order 6, and the maximum order? I'm not entirely sure where to go from here.

REPLY [5 votes]: Since it's been proposed that a similar question be closed as a duplicate of this one, I'll derive the number of permutations corresponding to a given cycle type in more generality.
Consider a cycle structure with $j_k$ cycles of length $k$, with $\sum_kkj_k=n$, and imagine it written out as pairs of parentheses enclosing appropriate numbers of slots. Each of the $n!$ assignments of the numbers from $1$ to $n$ to the slots yields a permutation, and we want to know how many are distinct.
Within each cycle of length $k$, there are $k$ cyclic permutations of the entries that would yield the same cycle. Among $j_k$ cycles of length $k$, there are $j_k!$ permutations that merely rearrange cycles of the same length. Thus $\prod_kk^{j_k}j_k!$ assignments lead to the same permutation, so the number of distinct permutations is
$$\frac{n!}{\prod_kk^{j_k}j_k!}\;.$$
See also this Wikipedia section.