On the proportion of elements of prime order in finite symmetric groups

We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order p, acting on a set of given size n, which is sharp for certain n and p. Namely, we prove that if n\equiv k\pmod{p} with ۰\leq k\leq p-۱, then this proportion is at most (p\cdot k!)^{-۱} with equality if and only if p\leq n<۲n.

Keywords:

finite symmetric groups , element proportions , elements of prime order