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

TL;DR

This paper investigates the distribution of element orders in Sylow p-subgroups of symmetric groups, revealing that the logarithm of element order concentrates around a specific growth rate with bounded variance.

Contribution

It introduces a probabilistic analysis of element orders in Sylow p-subgroups, showing their mean grows as (log n)/ (log p) with bounded variance, highlighting a difference from uniform element distributions.

Findings

Mean order of elements grows as (log n)/ (log p)

Variance of the logarithm of element order is bounded

Distribution differs from uniform element selection

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.