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 $n\geq 8$.
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