Most permutations power to a cycle of small prime length

TL;DR

This paper proves that as permutations grow large, most have a power forming a prime-length cycle around log n, with explicit bounds on the proportion of such permutations, including even permutations.

Contribution

It establishes the asymptotic density of permutations with prime-length cycle powers, providing explicit bounds and extending understanding of permutation cycle structures.

Findings

Most permutations have a power that is a prime-length cycle.

The proportion of such permutations approaches 1 as n increases.

Explicit bounds are given for the prime length and permutation parity.

Abstract

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log\log n$ with the prime between $\log n$ and $(\log n)^{\log\log n}$. The proportion of even permutations with this property is at least $1-7/\log\log n$.