Powers in wreath products of finite groups

Abstract

In this paper, we compute powers in the wreath product G≀SnG≀Sn for any finite group 𝐺. For r≥2r≥2 a prime, consider ωr:G≀Sn→G≀Snωr:G≀Sn→G≀Sn defined by g↦grg↦gr . Let Pr(G≀Sn):=|ωr(G≀Sn)G|nn!Pr⁢(G≀Sn):=|ωr⁢(G≀Sn)G|n⁢n! be the probability that a randomly chosen element in G≀SnG≀Sn is an 𝑟-th power. We prove Pr(G≀Sn+1)=Pr(G≀Sn)Pr⁢(G≀Sn+1)=Pr⁢(G≀Sn) for all n≢−1(modr)n≢-1⁢(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≀SnG≀Sn .