Gelfand pairs involving the wreath product of finite abelian groups with symmetric groups

Abstract: It is well known that the pair $(\mathcal{S}n,\mathcal{S}{n-1})$ is a Gelfand pair where $\mathcal{S}n$ is the symmetric group on $n$ elements. In this paper, we prove that if $G$ is a finite group then $(G\wr \mathcal{S}_n, G\wr \mathcal{S}{n-1}),$ where $G\wr \mathcal{S}_n$ is the wreath product of $G$ by $\mathcal{S}_n,$ is a Gelfand pair if and only if $G$ is abelian.