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 ≡ k (mod p) with 0 ≤ k ≤ p − 1, then this proportion is at most (p · k!)⁻¹ with equality if and only if p ≤ n < 2n.