The order of elements in Sylow $p$-subgroups of the symmetric group

Abstract

Define a random variable $\xi_n$ by choosing a conjugacy class $C$ of the Sylow $p$-subgroup of $S_{p^n}$ by random, and let $\xi_n$ be the logarithm of the order of an element in $C$. We show that $\xi_n$ has bounded variance and mean order $\frac{\log n}{\log p}+O(1)$, which differs significantly from the average order of elements chosen with equal probability.