Powers in wreath products of finite groups (Q2170272)

Let \(\Gamma\) be a finite group, \(r \in \mathbb{N}\) with \(r \geq 2\) and \(\omega_{r}(\Gamma)=\{\gamma^{r} \mid \gamma \in \Gamma\}\). The probability that a randomly chosen element in \(\Gamma\) is an \(r\)-th power is \(P_{r}(\Gamma)=|\omega_{r}(\Gamma)| \cdot |\Gamma|^{-1}\). The paper under review is devoted to computing \(P_{r}(G \wr S_{n})\) in the case where \(r\) is a prime. If \((|G|,r)=1\), then the authors prove that \(P_{r}(G \wr S_{n+1})=P_{r}(G \wr S_{n})\) for all \(n \not \equiv -1 \mod r\) (Theorem 5.6). They also provide a formula for the number of conjugacy classes that are \(r\)-th powers in \(G\wr S_{n}\).