Derangements in Sylow Subgroups of Symmetric Groups

Abstract

Let $H_n<S_n$, where $H_n$ is a Sylow $p$-subgroup of $S_n$, the symmetric group on $n$ letters. Let $A_n$ denote the number of derangements in $H_n$, and $f_n=\frac{h_n}{|H_n|}$. We will show that the sequence $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ is dense in the unit interval, but is Cesàro convergent to $0$.