Permutations with orders coprime to a given integer

Abstract

Let $m$ be a positive integer and let $ρ(m,n)$ be the proportion of permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $ρ(n,m)n^{1-\frac{ϕ(m)}{m}}\sim κ_m$ where $κ_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geqslant m$, \[C(m) \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1} \leqslant ρ(n,m) \leqslant \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1}\] where $ϕ$ is Euler's totient function.

10 pages, 3 figures