Derangements in wreath products of permutation 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 $\mathrm{A}_m^{[n]}$.