Derangements in wreath products of permutation groups

TL;DR

This paper derives formulas for fixed point proportions in wreath products of permutation groups and shows their density properties, providing new insights into the structure of derangements in these complex group actions.

Contribution

The paper introduces explicit formulas for fixed point proportions in wreath products and demonstrates their density, advancing understanding of derangements in permutation group constructions.

Findings

Formulas for fixed point proportions in wreath products

Density results of derangement proportions in certain group actions

Estimates for fixed point proportions in subgroups containing alternating groups

Abstract

Given a finite group $G$ acting on a set $X$ let $\delta_k(G,X)$ denote the proportion of elements in $G$ that have exactly $k$ fixed points in $X$. Let $\mathrm{S}_n$ denote the symmetric group acting on $[n]=\{1,2,\dots,n\}$. For $A\le\mathrm{S}_m$ and $B\le\mathrm{S}_n$, the permutational wreath product $A\wr B$ has two natural actions and we give formulas for both, $\delta_k(A\wr B,[m]{\times}[n])$ and $\delta_k(A\wr B,[m]^{[n]})$. We prove that for $k=0$ the values of these proportions are dense in the intervals $[\delta_0(B,[n]),1]$ and $[\delta_0(A,[m]),1]$. Among further result, we provide estimates for $\delta_0(G,[m]^{[n]})$ for subgroups $G\leq \mathrm{S}_m\wr\mathrm{S}_n$ containing $\mathrm{A}_m^{[n]}$.