Powers in the wreath product of $G$ with $S_n$

TL;DR

This paper calculates the distribution of $r^{th}$ powers in wreath products of finite groups with symmetric groups, providing formulas and invariance properties under certain conditions.

Contribution

It derives formulas for the number of conjugacy classes that are $r^{th}$ powers and shows invariance of the probability of $r^{th}$ powers across certain group sizes.

Findings

Probability of $r^{th}$ powers remains constant for specific $n$ when $ ext{gcd}(|G|, r)=1$.

Provides explicit formulas for counting $r^{th}$ power conjugacy classes in $G \, ext{wr}\, S_n$.

Establishes conditions under which the distribution of $r^{th}$ powers is invariant.

Abstract

In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $\omega_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|\omega_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.