Powers in wreath products of finite groups

Abstract

Abstract In this paper, we compute powers in the wreath product G ≀ S n G\wr S_{n} for any finite group 𝐺. For r ≥ 2 r\geq 2 a prime, consider ω r : G ≀ S n → G ≀ S n \omega_{r}\colon G\wr S_{n}\to G\wr S_{n} defined by g ↦ g r g\mapsto g^{r} . Let P r ⁢ ( G ≀ S n ) := | ω r ⁢ ( G ≀ S n ) | | G | n ⁢ n ! P_{r}(G\wr S_{n}):=\frac{\lvert\omega_{r}(G\wr S_{n})\rvert}{\lvert G\rvert^{n}n!} be the probability that a randomly chosen element in G ≀ S n G\wr S_{n} is an 𝑟-th power. We prove P r ⁢ ( G ≀ S n + 1 ) = P r ⁢ ( G ≀ S n ) P_{r}(G\wr S_{n+1})=P_{r}(G\wr S_{n}) for all n ≢ - 1 ⁢ ( mod ⁢ r ) n\not\equiv-1\ (\mathrm{mod}\ r) if the 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 ≀ S n G\wr S_{n} .