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In this paper we compute powers in the wreath product $G\wr S_n$, for anyfinite group $G$. For $r\geq 2$, a prime, consider $\omega_r: G\wr S_n\to G\wrS_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|\omega_r(G\wrS_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wrS_n$ 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 alsogive a formula for the number of conjugacy classes that are $r^{th}$ powers in$G\wr S_n$.

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