On the proportion of elements of prime order in finite symmetric groups

Document Type : 2022 CCGTA IN SOUTH FLA


1 Centre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth 6009, Australia

2 Department of Mathematics, Federal University Gashua, Yobe State, Nigeria


We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n<2n$.


Main Subjects

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