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

TL;DR

This paper provides a concise proof establishing an explicit upper bound on the proportion of permutations of a specific prime order in finite symmetric groups, with conditions for equality based on the size of the set and the prime.

Contribution

The paper introduces a new, simplified proof for an explicit upper bound on the proportion of prime order permutations in symmetric groups, clarifying the conditions for equality.

Findings

Upper bound of (p * k!)^{-1} on permutation proportion

Equality holds if and only if p ≤ n < 2n

Bound is sharp for certain n and p

Abstract

We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a finite set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n<2n$.