A conjecture of Glasby, Praeger, and Unger on permutations of $S_n$

Abstract

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_{n}$. Let $π_{n}$ denote the proportion of elements of $S_{n}$ that are pre-$p$-cycles for some prime $p\in[2, n-3]$. We prove that $π_{n} > 1/3$ for all $n\geq 8$.