A family of finite Gelfand pairs associated with wreath products
Consider the wreath product $G_n=\varGamma ^n\rtimes S_n$ of a finite group $\varGamma $ with the symmetric group $S_n$. Let $\varDelta _n$ denote the diagonal in $\varGamma ^n$. Then $K_n=\varDelta _n\times S_n$ forms a subgroup of $G_n$. In case $\varGamma $ is abelian, $(G_n,K_n)$ forms a Gelfand pair with relevance to the study of parking functions. For $\varGamma $ non-abelian it was suggested by Kürşat Aker and Mahir Bilen Can that $(G_n,K_n)$ must fail to be a Gelfand pair for $n$ sufficiently large. We prove that this is indeed the case: for $\varGamma $ non-abelian there is some integer $2 \lt N(\varGamma )\le |\varGamma |$ for which $(K_n,G_n)$ is a Gelfand pair for all $n \lt N(\varGamma )$ but $(K_n,G_n)$ fails to be a Gelfand pair for all $n\ge N(\varGamma )$. For dihedral groups $\varGamma =D_p$ with $p$ an odd prime we prove that $N(\varGamma )=6$.