Published online by Cambridge University Press: 13 April 2020
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.