论文标题
置换组的花圈产品中的危险
Derangements in wreath products of permutation groups
论文作者
论文摘要
鉴于有限的组$ g $在套装$ x $中代表$δ_k（g，x）$表示$ g $中具有$ k $固定点$ x $的元素的比例。令$ \ mathrm {s} _n $表示对称组，该组作用于$ [n] = \ {1,2，\ dots，n \} $。对于$ a \ le \ mathrm {s} _m $和$ b \ le \ mathrm {s} _n $，排列的花环产品$ a \ wr b $具有两个自然动作，我们给出了两者的公式，$Δ_K（$Δ_K（a \ wr b，a \ wr b，[m] {m] {\ times} {\ n n n n n n]和$Δ__ b，[m]^{[n]}）$。我们证明，对于$ k = 0 $，这些比例的值在$ [Δ_0（b，[n]），1] $和$ [δ_0（a，[m]），1] $的间隔中密集。在进一步的结果中，我们提供了$Δ_0（g，[m]^{[n]}）$的估计值，用于子组$ g \ leq \ leq \ mathrm {s} _m \ wr \ mathrm {s} s} _n $ conting $
Given a finite group $G$ acting on a set $X$ let $δ_k(G,X)$ 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, $δ_k(A\wr B,[m]{\times}[n])$ and $δ_k(A\wr B,[m]^{[n]})$. We prove that for $k=0$ the values of these proportions are dense in the intervals $[δ_0(B,[n]),1]$ and $[δ_0(A,[m]),1]$. Among further result, we provide estimates for $δ_0(G,[m]^{[n]})$ for subgroups $G\leq \mathrm{S}_m\wr\mathrm{S}_n$ containing $\mathrm{A}_m^{[n]}$.
