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arXiv · 2107.02724

The proportion of derangements characterizes the symmetric and alternating groups

Bjorn Poonen·Kaloyan Slavov
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Abstract

Let $G$ be a subgroup of the symmetric group $S_n$. If the proportion of fixed-point-free elements in $G$ (or a coset) equals the proportion of fixed-point-free elements in $S_n$, then $G=S_n$. The analogue for $A_n$ holds if $n \ge 7$. We give an application to monodromy groups.

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Bjorn Poonen, Kaloyan Slavov. 2021-07-06. The proportion of derangements characterizes the symmetric and alternating groups. https://doi.org/10.1112/blms.12639

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