The proportion of derangements characterizes the symmetric and alternating groups

TL;DR

This paper characterizes symmetric and alternating groups by the proportion of fixed-point-free elements, providing a new criterion for identifying these groups and applying it to monodromy groups.

Contribution

It establishes a novel characterization of symmetric and alternating groups based on fixed-point-free element proportions, with implications for monodromy group analysis.

Findings

Proportion of fixed-point-free elements uniquely identifies $S_n$ and $A_n$ under certain conditions.

Provides a new criterion for group identification in permutation groups.

Applications to monodromy groups demonstrate practical relevance.

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.